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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St001902
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Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 0
Description
The number of potential covers of a poset.
A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations.
For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.
Matching statistic: St001635
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> -1 = 0 - 1
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
([(0,5),(1,4),(1,5),(4,3),(4,6),(5,6),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 1
([(1,6),(2,3),(2,6),(3,5),(6,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,6),(1,4),(1,6),(3,5),(4,3),(6,2),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 1
([(0,6),(1,3),(1,6),(3,4),(4,2),(4,5),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 1
([(0,5),(0,6),(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,4),(1,5),(1,6),(4,3),(5,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 3 - 1
([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 4 - 1
([(0,4),(1,3),(1,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 1
([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 3 - 1
([(0,3),(1,4),(1,6),(3,6),(4,5),(6,2),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 1
([(0,3),(1,2),(1,6),(2,4),(2,5),(3,4),(3,6),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 1
([(0,4),(1,5),(1,6),(5,3),(6,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
([(0,5),(0,6),(1,2),(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2 - 1
([(0,3),(0,6),(1,5),(1,6),(3,4),(3,5),(5,2),(6,4)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2 - 1
([(0,5),(0,6),(1,4),(1,6),(3,5),(4,3),(6,2)],7)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,5),(3,6),(4,3),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(3,6),(4,5),(6,2)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 4 - 1
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 4 - 1
([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,5),(0,6),(1,4),(3,5),(4,3),(4,6),(6,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2 - 1
([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 3 - 1
([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 4 - 1
([(0,6),(1,5),(4,2),(5,4),(5,6),(6,3)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2 - 1
([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
([(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
([(0,6),(1,5),(2,3),(2,5),(2,6),(3,4)],7)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 6 - 1
([(0,6),(1,5),(2,4),(2,5),(2,6),(6,3)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5 - 1
([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 3 - 1
Description
The trace of the square of the Coxeter matrix of the incidence algebra of a poset.
Matching statistic: St000311
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ? = 4
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,5),(1,4),(1,5),(4,3),(4,6),(5,6),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(6,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2
([(0,6),(1,4),(1,6),(3,5),(4,3),(6,2),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(1,6),(3,4),(4,2),(4,5),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,6),(1,4),(1,5),(1,6),(4,3),(5,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 3
([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(4,7),(5,6)],8)
=> ? = 4
([(0,4),(1,3),(1,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> ? = 2
([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,5),(6,2),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> ? = 2
([(0,3),(1,2),(1,6),(2,4),(2,5),(3,4),(3,6),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> ? = 2
([(0,4),(1,5),(1,6),(5,3),(6,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,5),(0,6),(1,2),(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,3),(0,6),(1,5),(1,6),(3,5),(5,4),(6,2),(6,4)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ? = 2
([(0,3),(0,6),(1,5),(1,6),(3,4),(3,5),(5,2),(6,4)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(3,5),(4,3),(6,2)],7)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 3
([(0,5),(0,6),(1,4),(1,5),(3,6),(4,3),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ? = 2
([(1,4),(1,6),(2,3),(2,5),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ? = 4
([(0,4),(0,6),(1,3),(1,5),(3,6),(4,5),(6,2)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> ? = 4
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,5),(5,4),(6,4)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ? = 4
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(4,7),(5,6)],8)
=> ? = 4
([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(0,6),(1,4),(3,5),(4,3),(4,6),(6,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 2
([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3
([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> ? = 4
([(0,6),(1,5),(4,2),(5,4),(5,6),(6,3)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(3,4),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ? = 2
([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
Description
The number of vertices of odd degree in a graph.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,3),(0,4),(0,5),(3,6),(4,2),(5,1),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(6,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2
([(0,6),(1,4),(1,6),(3,5),(4,3),(6,2),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(1,6),(3,4),(4,2),(4,5),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(6,3)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,6),(1,3),(1,5),(1,6),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,6),(1,4),(1,5),(1,6),(4,3),(5,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,6),(1,3),(1,4),(1,6),(3,5),(4,2),(4,5)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,3),(0,5),(0,6),(4,1),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(1,3),(1,5),(1,6),(4,6),(5,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 3
([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(4,7),(5,6)],8)
=> ? = 4
([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 3
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(1,5),(1,6),(5,3),(6,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,6),(1,4),(1,5),(4,3),(5,2),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,2),(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(3,5),(4,3),(6,2)],7)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 3
([(0,5),(0,6),(1,3),(1,4),(1,6),(3,5),(4,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(0,6),(1,3),(1,5),(3,6),(4,5),(6,2)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> ? = 4
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(4,7),(5,6)],8)
=> ? = 4
([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001618
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 4
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 3
([(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 2
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,2),(2,6),(2,7),(2,8),(3,17),(4,16),(5,15),(6,12),(6,13),(7,12),(7,14),(8,13),(8,14),(9,19),(10,19),(11,19),(12,5),(12,18),(13,4),(13,18),(14,3),(14,18),(15,9),(15,10),(16,9),(16,11),(17,10),(17,11),(18,15),(18,16),(18,17),(19,1)],20)
=> ? = 6
([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2
([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,2),(2,6),(2,7),(2,8),(3,17),(4,16),(5,15),(6,12),(6,13),(7,12),(7,14),(8,13),(8,14),(9,19),(10,19),(11,19),(12,5),(12,18),(13,4),(13,18),(14,3),(14,18),(15,9),(15,10),(16,9),(16,11),(17,10),(17,11),(18,15),(18,16),(18,17),(19,1)],20)
=> ? = 6
([(0,3),(0,4),(0,5),(3,6),(4,2),(5,1),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,5),(1,4),(1,5),(4,3),(4,6),(5,6),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(6,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 2
([(0,6),(1,4),(1,6),(3,5),(4,3),(6,2),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2
([(0,6),(1,3),(1,6),(3,4),(4,2),(4,5),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(6,3)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,6),(1,3),(1,5),(1,6),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,6),(1,4),(1,5),(1,6),(4,3),(5,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,2),(2,6),(2,7),(2,8),(3,17),(4,16),(5,15),(6,12),(6,13),(7,12),(7,14),(8,13),(8,14),(9,19),(10,19),(11,19),(12,5),(12,18),(13,4),(13,18),(14,3),(14,18),(15,9),(15,10),(16,9),(16,11),(17,10),(17,11),(18,15),(18,16),(18,17),(19,1)],20)
=> ? = 6
([(0,6),(1,3),(1,4),(1,6),(3,5),(4,2),(4,5)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,3),(0,5),(0,6),(4,1),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(1,3),(1,5),(1,6),(4,6),(5,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 3
([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,8),(2,10),(3,10),(4,9),(5,9),(6,7),(7,2),(7,3),(8,4),(8,5),(9,6),(10,1)],11)
=> ? = 4
([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 3
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(1,5),(1,6),(5,3),(6,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,2),(2,6),(2,7),(2,8),(3,17),(4,16),(5,15),(6,12),(6,13),(7,12),(7,14),(8,13),(8,14),(9,19),(10,19),(11,19),(12,5),(12,18),(13,4),(13,18),(14,3),(14,18),(15,9),(15,10),(16,9),(16,11),(17,10),(17,11),(18,15),(18,16),(18,17),(19,1)],20)
=> ? = 6
([(0,6),(1,4),(1,5),(4,3),(5,2),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,5),(0,6),(1,2),(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2
([(0,3),(0,6),(1,5),(1,6),(3,4),(3,5),(5,2),(6,4)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(3,5),(4,3),(6,2)],7)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 3
([(0,5),(0,6),(1,4),(1,5),(3,6),(4,3),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 2
([(0,5),(0,6),(1,3),(1,4),(1,6),(3,5),(4,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
([(1,4),(1,6),(2,3),(2,5),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 4
([(0,4),(0,6),(1,3),(1,5),(3,6),(4,5),(6,2)],7)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 4
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,5),(5,4),(6,4)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 4
([(0,3),(0,6),(1,2),(1,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,8),(2,9),(3,9),(4,10),(5,10),(6,1),(7,4),(7,5),(8,2),(8,3),(9,7),(10,6)],11)
=> ? = 4
Description
The cardinality of the Frattini sublattice of a lattice.
The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.
Matching statistic: St001703
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ? = 4
([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(1,2),(1,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 2
([(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0
([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,3),(0,4),(0,5),(3,6),(4,2),(5,1),(5,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(1,4),(1,5),(4,3),(4,6),(5,6),(6,2)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(6,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2
([(0,6),(1,3),(1,6),(3,5),(5,4),(6,2),(6,5)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0
([(0,6),(1,4),(1,6),(3,5),(4,3),(6,2),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(1,6),(3,4),(4,2),(4,5),(6,5)],7)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,6),(1,2),(1,6),(2,3),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(5,4),(6,2),(6,4)],7)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(3,4),(5,3),(6,2),(6,4)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(6,3)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,6),(1,3),(1,5),(1,6),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,6),(1,4),(1,5),(1,6),(4,3),(5,2)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 6
([(0,6),(1,3),(1,4),(1,6),(3,5),(4,2),(4,5)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,3),(0,5),(0,6),(4,1),(5,2),(6,4)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,4),(1,3),(1,5),(1,6),(4,6),(5,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2
([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 2
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 2
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 3
([(0,2),(1,3),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 2
Description
The villainy of a graph.
The villainy of a permutation of a proper coloring $c$ of a graph is the minimal Hamming distance between $c$ and a proper coloring.
The villainy of a graph is the maximal villainy of a permutation of a proper coloring.
Matching statistic: St001651
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? = 0
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 2
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,16),(1,21),(1,22),(1,28),(1,47),(1,52),(1,53),(1,88),(1,89),(1,91),(2,15),(2,19),(2,20),(2,27),(2,46),(2,50),(2,51),(2,86),(2,87),(2,91),(3,18),(3,24),(3,26),(3,30),(3,49),(3,55),(3,57),(3,87),(3,89),(3,90),(4,17),(4,23),(4,25),(4,29),(4,48),(4,54),(4,56),(4,86),(4,88),(4,90),(5,15),(5,31),(5,32),(5,39),(5,47),(5,54),(5,55),(5,92),(5,93),(5,97),(6,16),(6,33),(6,34),(6,40),(6,46),(6,56),(6,57),(6,94),(6,95),(6,97),(7,17),(7,35),(7,37),(7,41),(7,49),(7,50),(7,52),(7,92),(7,94),(7,96),(8,18),(8,36),(8,38),(8,42),(8,48),(8,51),(8,53),(8,93),(8,95),(8,96),(9,19),(9,23),(9,31),(9,35),(9,44),(9,82),(9,84),(9,89),(9,95),(10,20),(10,24),(10,32),(10,36),(10,45),(10,82),(10,85),(10,88),(10,94),(11,21),(11,25),(11,33),(11,37),(11,45),(11,83),(11,84),(11,87),(11,93),(12,22),(12,26),(12,34),(12,38),(12,44),(12,83),(12,85),(12,86),(12,92),(13,29),(13,30),(13,41),(13,42),(13,43),(13,84),(13,85),(13,91),(13,97),(14,27),(14,28),(14,39),(14,40),(14,43),(14,82),(14,83),(14,90),(14,96),(15,70),(15,71),(15,78),(15,152),(15,153),(15,157),(16,72),(16,73),(16,79),(16,154),(16,155),(16,157),(17,74),(17,76),(17,80),(17,152),(17,154),(17,156),(18,75),(18,77),(18,81),(18,153),(18,155),(18,156),(19,58),(19,70),(19,99),(19,122),(19,134),(19,168),(20,59),(20,71),(20,98),(20,123),(20,134),(20,167),(21,60),(21,72),(21,101),(21,124),(21,135),(21,168),(22,61),(22,73),(22,100),(22,125),(22,135),(22,167),(23,62),(23,74),(23,104),(23,122),(23,136),(23,166),(24,63),(24,75),(24,105),(24,123),(24,137),(24,166),(25,64),(25,76),(25,102),(25,124),(25,136),(25,165),(26,65),(26,77),(26,103),(26,125),(26,137),(26,165),(27,66),(27,78),(27,106),(27,126),(27,134),(27,165),(28,67),(28,79),(28,107),(28,126),(28,135),(28,166),(29,68),(29,80),(29,108),(29,127),(29,136),(29,167),(30,69),(30,81),(30,109),(30,127),(30,137),(30,168),(31,62),(31,70),(31,113),(31,128),(31,138),(31,172),(32,63),(32,71),(32,112),(32,129),(32,138),(32,171),(33,64),(33,72),(33,111),(33,130),(33,139),(33,172),(34,65),(34,73),(34,110),(34,131),(34,139),(34,171),(35,58),(35,74),(35,117),(35,128),(35,140),(35,170),(36,59),(36,75),(36,116),(36,129),(36,141),(36,170),(37,60),(37,76),(37,115),(37,130),(37,140),(37,169),(38,61),(38,77),(38,114),(38,131),(38,141),(38,169),(39,67),(39,78),(39,118),(39,132),(39,138),(39,169),(40,66),(40,79),(40,119),(40,132),(40,139),(40,170),(41,69),(41,80),(41,120),(41,133),(41,140),(41,171),(42,68),(42,81),(42,121),(42,133),(42,141),(42,172),(43,126),(43,127),(43,132),(43,133),(43,158),(44,122),(44,125),(44,128),(44,131),(44,158),(45,123),(45,124),(45,129),(45,130),(45,158),(46,66),(46,98),(46,99),(46,110),(46,111),(46,157),(47,67),(47,100),(47,101),(47,112),(47,113),(47,157),(48,68),(48,102),(48,104),(48,114),(48,116),(48,156),(49,69),(49,103),(49,105),(49,115),(49,117),(49,156),(50,58),(50,98),(50,106),(50,115),(50,120),(50,152),(51,59),(51,99),(51,106),(51,114),(51,121),(51,153),(52,60),(52,100),(52,107),(52,117),(52,120),(52,154),(53,61),(53,101),(53,107),(53,116),(53,121),(53,155),(54,62),(54,102),(54,108),(54,112),(54,118),(54,152),(55,63),(55,103),(55,109),(55,113),(55,118),(55,153),(56,64),(56,104),(56,108),(56,110),(56,119),(56,154),(57,65),(57,105),(57,109),(57,111),(57,119),(57,155),(58,159),(58,173),(58,180),(59,160),(59,173),(59,179),(60,161),(60,174),(60,180),(61,162),(61,174),(61,179),(62,159),(62,175),(62,178),(63,160),(63,176),(63,178),(64,161),(64,175),(64,177),(65,162),(65,176),(65,177),(66,163),(66,173),(66,177),(67,163),(67,174),(67,178),(68,164),(68,175),(68,179),(69,164),(69,176),(69,180),(70,142),(70,159),(70,184),(71,142),(71,160),(71,183),(72,143),(72,161),(72,184),(73,143),(73,162),(73,183),(74,144),(74,159),(74,182),(75,145),(75,160),(75,182),(76,144),(76,161),(76,181),(77,145),(77,162),(77,181),(78,142),(78,163),(78,181),(79,143),(79,163),(79,182),(80,144),(80,164),(80,183),(81,145),(81,164),(81,184),(82,134),(82,138),(82,158),(82,166),(82,170),(83,135),(83,139),(83,158),(83,165),(83,169),(84,136),(84,140),(84,158),(84,168),(84,172),(85,137),(85,141),(85,158),(85,167),(85,171),(86,110),(86,114),(86,122),(86,152),(86,165),(86,167),(87,111),(87,115),(87,123),(87,153),(87,165),(87,168),(88,112),(88,116),(88,124),(88,154),(88,166),(88,167),(89,113),(89,117),(89,125),(89,155),(89,166),(89,168),(90,118),(90,119),(90,127),(90,156),(90,165),(90,166),(91,120),(91,121),(91,126),(91,157),(91,167),(91,168),(92,100),(92,103),(92,128),(92,152),(92,169),(92,171),(93,101),(93,102),(93,129),(93,153),(93,169),(93,172),(94,98),(94,105),(94,130),(94,154),(94,170),(94,171),(95,99),(95,104),(95,131),(95,155),(95,170),(95,172),(96,106),(96,107),(96,133),(96,156),(96,169),(96,170),(97,108),(97,109),(97,132),(97,157),(97,171),(97,172),(98,147),(98,173),(98,183),(99,146),(99,173),(99,184),(100,149),(100,174),(100,183),(101,148),(101,174),(101,184),(102,148),(102,175),(102,181),(103,149),(103,176),(103,181),(104,146),(104,175),(104,182),(105,147),(105,176),(105,182),(106,150),(106,173),(106,181),(107,150),(107,174),(107,182),(108,151),(108,175),(108,183),(109,151),(109,176),(109,184),(110,146),(110,177),(110,183),(111,147),(111,177),(111,184),(112,148),(112,178),(112,183),(113,149),(113,178),(113,184),(114,146),(114,179),(114,181),(115,147),(115,180),(115,181),(116,148),(116,179),(116,182),(117,149),(117,180),(117,182),(118,151),(118,178),(118,181),(119,151),(119,177),(119,182),(120,150),(120,180),(120,183),(121,150),(121,179),(121,184),(122,146),(122,159),(122,186),(123,147),(123,160),(123,186),(124,148),(124,161),(124,186),(125,149),(125,162),(125,186),(126,150),(126,163),(126,186),(127,151),(127,164),(127,186),(128,149),(128,159),(128,185),(129,148),(129,160),(129,185),(130,147),(130,161),(130,185),(131,146),(131,162),(131,185),(132,151),(132,163),(132,185),(133,150),(133,164),(133,185),(134,142),(134,173),(134,186),(135,143),(135,174),(135,186),(136,144),(136,175),(136,186),(137,145),(137,176),(137,186),(138,142),(138,178),(138,185),(139,143),(139,177),(139,185),(140,144),(140,180),(140,185),(141,145),(141,179),(141,185),(142,187),(143,187),(144,187),(145,187),(146,187),(147,187),(148,187),(149,187),(150,187),(151,187),(152,159),(152,181),(152,183),(153,160),(153,181),(153,184),(154,161),(154,182),(154,183),(155,162),(155,182),(155,184),(156,164),(156,181),(156,182),(157,163),(157,183),(157,184),(158,185),(158,186),(159,187),(160,187),(161,187),(162,187),(163,187),(164,187),(165,177),(165,181),(165,186),(166,178),(166,182),(166,186),(167,179),(167,183),(167,186),(168,180),(168,184),(168,186),(169,174),(169,181),(169,185),(170,173),(170,182),(170,185),(171,176),(171,183),(171,185),(172,175),(172,184),(172,185),(173,187),(174,187),(175,187),(176,187),(177,187),(178,187),(179,187),(180,187),(181,187),(182,187),(183,187),(184,187),(185,187),(186,187)],188)
=> ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 0
([(2,3),(2,4),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,17),(1,30),(1,31),(1,39),(1,40),(1,41),(1,42),(1,49),(2,14),(2,15),(2,28),(2,29),(2,35),(2,36),(2,37),(2,38),(2,49),(3,19),(3,23),(3,27),(3,34),(3,36),(3,40),(3,74),(3,79),(4,18),(4,22),(4,26),(4,33),(4,35),(4,39),(4,74),(4,78),(5,21),(5,22),(5,24),(5,34),(5,37),(5,41),(5,75),(5,76),(6,20),(6,23),(6,25),(6,33),(6,38),(6,42),(6,75),(6,77),(7,13),(7,20),(7,21),(7,29),(7,31),(7,32),(7,78),(7,79),(8,12),(8,18),(8,19),(8,28),(8,30),(8,32),(8,76),(8,77),(9,14),(9,16),(9,25),(9,26),(9,48),(9,76),(9,79),(10,15),(10,17),(10,24),(10,27),(10,48),(10,77),(10,78),(11,12),(11,13),(11,48),(11,49),(11,74),(11,75),(12,43),(12,84),(12,98),(12,121),(13,43),(13,85),(13,99),(13,122),(14,55),(14,56),(14,86),(14,89),(14,109),(15,54),(15,57),(15,87),(15,88),(15,109),(16,59),(16,60),(16,90),(16,93),(16,109),(17,58),(17,61),(17,91),(17,92),(17,109),(18,62),(18,66),(18,84),(18,94),(18,113),(19,63),(19,67),(19,84),(19,95),(19,114),(20,64),(20,68),(20,85),(20,97),(20,113),(21,65),(21,69),(21,85),(21,96),(21,114),(22,50),(22,52),(22,94),(22,96),(22,110),(23,51),(23,53),(23,95),(23,97),(23,110),(24,54),(24,58),(24,71),(24,96),(24,121),(25,55),(25,59),(25,70),(25,97),(25,121),(26,56),(26,60),(26,70),(26,94),(26,122),(27,57),(27,61),(27,71),(27,95),(27,122),(28,62),(28,63),(28,72),(28,86),(28,87),(28,98),(29,64),(29,65),(29,72),(29,88),(29,89),(29,99),(30,66),(30,67),(30,73),(30,90),(30,91),(30,98),(31,68),(31,69),(31,73),(31,92),(31,93),(31,99),(32,43),(32,72),(32,73),(32,113),(32,114),(33,44),(33,46),(33,70),(33,110),(33,113),(34,45),(34,47),(34,71),(34,110),(34,114),(35,44),(35,50),(35,56),(35,62),(35,88),(35,111),(36,45),(36,51),(36,57),(36,63),(36,89),(36,111),(37,45),(37,50),(37,54),(37,65),(37,86),(37,112),(38,44),(38,51),(38,55),(38,64),(38,87),(38,112),(39,46),(39,52),(39,60),(39,66),(39,92),(39,111),(40,47),(40,53),(40,61),(40,67),(40,93),(40,111),(41,47),(41,52),(41,58),(41,69),(41,90),(41,112),(42,46),(42,53),(42,59),(42,68),(42,91),(42,112),(43,100),(43,126),(44,80),(44,117),(44,123),(45,81),(45,118),(45,123),(46,82),(46,119),(46,123),(47,83),(47,120),(47,123),(48,109),(48,121),(48,122),(49,98),(49,99),(49,109),(49,111),(49,112),(50,101),(50,103),(50,123),(51,102),(51,104),(51,123),(52,105),(52,107),(52,123),(53,106),(53,108),(53,123),(54,81),(54,103),(54,124),(55,80),(55,104),(55,124),(56,80),(56,101),(56,125),(57,81),(57,102),(57,125),(58,83),(58,107),(58,124),(59,82),(59,108),(59,124),(60,82),(60,105),(60,125),(61,83),(61,106),(61,125),(62,101),(62,115),(62,117),(63,102),(63,115),(63,118),(64,104),(64,116),(64,117),(65,103),(65,116),(65,118),(66,105),(66,115),(66,119),(67,106),(67,115),(67,120),(68,108),(68,116),(68,119),(69,107),(69,116),(69,120),(70,80),(70,82),(70,126),(71,81),(71,83),(71,126),(72,100),(72,117),(72,118),(73,100),(73,119),(73,120),(74,84),(74,110),(74,111),(74,122),(75,85),(75,110),(75,112),(75,121),(76,86),(76,90),(76,94),(76,114),(76,121),(77,87),(77,91),(77,95),(77,113),(77,121),(78,88),(78,92),(78,96),(78,113),(78,122),(79,89),(79,93),(79,97),(79,114),(79,122),(80,127),(81,127),(82,127),(83,127),(84,115),(84,126),(85,116),(85,126),(86,101),(86,118),(86,124),(87,102),(87,117),(87,124),(88,103),(88,117),(88,125),(89,104),(89,118),(89,125),(90,105),(90,120),(90,124),(91,106),(91,119),(91,124),(92,107),(92,119),(92,125),(93,108),(93,120),(93,125),(94,101),(94,105),(94,126),(95,102),(95,106),(95,126),(96,103),(96,107),(96,126),(97,104),(97,108),(97,126),(98,100),(98,115),(98,124),(99,100),(99,116),(99,125),(100,127),(101,127),(102,127),(103,127),(104,127),(105,127),(106,127),(107,127),(108,127),(109,124),(109,125),(110,123),(110,126),(111,115),(111,123),(111,125),(112,116),(112,123),(112,124),(113,117),(113,119),(113,126),(114,118),(114,120),(114,126),(115,127),(116,127),(117,127),(118,127),(119,127),(120,127),(121,124),(121,126),(122,125),(122,126),(123,127),(124,127),(125,127),(126,127)],128)
=> ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,21),(1,25),(1,29),(1,33),(1,55),(1,57),(2,20),(2,24),(2,28),(2,32),(2,55),(2,56),(3,22),(3,26),(3,30),(3,32),(3,54),(3,57),(4,23),(4,27),(4,31),(4,33),(4,54),(4,56),(5,11),(5,14),(5,15),(5,17),(5,22),(5,23),(5,55),(6,11),(6,12),(6,13),(6,16),(6,20),(6,21),(6,54),(7,10),(7,12),(7,14),(7,18),(7,24),(7,27),(7,57),(8,10),(8,13),(8,15),(8,19),(8,25),(8,26),(8,56),(9,16),(9,17),(9,18),(9,19),(9,28),(9,29),(9,30),(9,31),(10,53),(10,60),(10,61),(10,74),(11,52),(11,58),(11,59),(11,74),(12,34),(12,38),(12,74),(12,76),(13,35),(13,39),(13,74),(13,75),(14,37),(14,40),(14,74),(14,77),(15,36),(15,41),(15,74),(15,78),(16,38),(16,39),(16,44),(16,45),(16,52),(16,62),(17,40),(17,41),(17,46),(17,47),(17,52),(17,63),(18,38),(18,40),(18,48),(18,51),(18,53),(18,65),(19,39),(19,41),(19,49),(19,50),(19,53),(19,64),(20,34),(20,44),(20,58),(20,75),(21,35),(21,45),(21,58),(21,76),(22,36),(22,46),(22,59),(22,77),(23,37),(23,47),(23,59),(23,78),(24,34),(24,48),(24,60),(24,77),(25,35),(25,49),(25,61),(25,78),(26,36),(26,50),(26,61),(26,75),(27,37),(27,51),(27,60),(27,76),(28,42),(28,44),(28,48),(28,63),(28,64),(29,43),(29,45),(29,49),(29,63),(29,65),(30,42),(30,46),(30,50),(30,62),(30,65),(31,43),(31,47),(31,51),(31,62),(31,64),(32,42),(32,75),(32,77),(33,43),(33,76),(33,78),(34,66),(34,84),(35,67),(35,84),(36,68),(36,84),(37,69),(37,84),(38,66),(38,80),(38,83),(39,67),(39,79),(39,83),(40,69),(40,81),(40,83),(41,68),(41,82),(41,83),(42,79),(42,81),(43,80),(43,82),(44,66),(44,70),(44,79),(45,67),(45,70),(45,80),(46,68),(46,71),(46,81),(47,69),(47,71),(47,82),(48,66),(48,72),(48,81),(49,67),(49,73),(49,82),(50,68),(50,73),(50,79),(51,69),(51,72),(51,80),(52,70),(52,71),(52,83),(53,72),(53,73),(53,83),(54,59),(54,62),(54,75),(54,76),(55,58),(55,63),(55,77),(55,78),(56,60),(56,64),(56,75),(56,78),(57,61),(57,65),(57,76),(57,77),(58,70),(58,84),(59,71),(59,84),(60,72),(60,84),(61,73),(61,84),(62,71),(62,79),(62,80),(63,70),(63,81),(63,82),(64,72),(64,79),(64,82),(65,73),(65,80),(65,81),(66,85),(67,85),(68,85),(69,85),(70,85),(71,85),(72,85),(73,85),(74,83),(74,84),(75,79),(75,84),(76,80),(76,84),(77,81),(77,84),(78,82),(78,84),(79,85),(80,85),(81,85),(82,85),(83,85),(84,85)],86)
=> ? = 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(1,13),(1,14),(1,44),(1,46),(2,9),(2,11),(2,12),(2,44),(2,45),(3,15),(3,24),(3,25),(3,26),(3,27),(3,44),(4,12),(4,17),(4,22),(4,23),(4,25),(4,46),(5,11),(5,16),(5,20),(5,21),(5,24),(5,46),(6,14),(6,19),(6,21),(6,23),(6,27),(6,45),(7,13),(7,18),(7,20),(7,22),(7,26),(7,45),(8,9),(8,10),(8,15),(8,16),(8,17),(8,18),(8,19),(9,36),(9,37),(9,47),(9,52),(10,38),(10,39),(10,47),(10,53),(11,36),(11,48),(11,62),(12,37),(12,49),(12,62),(13,38),(13,50),(13,63),(14,39),(14,51),(14,63),(15,32),(15,33),(15,34),(15,35),(15,47),(16,32),(16,36),(16,40),(16,41),(16,53),(17,33),(17,37),(17,42),(17,43),(17,53),(18,34),(18,38),(18,40),(18,42),(18,52),(19,35),(19,39),(19,41),(19,43),(19,52),(20,28),(20,40),(20,48),(20,50),(21,29),(21,41),(21,48),(21,51),(22,30),(22,42),(22,49),(22,50),(23,31),(23,43),(23,49),(23,51),(24,28),(24,29),(24,32),(24,62),(25,30),(25,31),(25,33),(25,62),(26,28),(26,30),(26,34),(26,63),(27,29),(27,31),(27,35),(27,63),(28,54),(28,66),(29,55),(29,66),(30,56),(30,66),(31,57),(31,66),(32,54),(32,55),(32,64),(33,56),(33,57),(33,64),(34,54),(34,56),(34,65),(35,55),(35,57),(35,65),(36,58),(36,64),(37,59),(37,64),(38,60),(38,65),(39,61),(39,65),(40,54),(40,58),(40,60),(41,55),(41,58),(41,61),(42,56),(42,59),(42,60),(43,57),(43,59),(43,61),(44,47),(44,62),(44,63),(45,48),(45,49),(45,52),(45,63),(46,50),(46,51),(46,53),(46,62),(47,64),(47,65),(48,58),(48,66),(49,59),(49,66),(50,60),(50,66),(51,61),(51,66),(52,58),(52,59),(52,65),(53,60),(53,61),(53,64),(54,67),(55,67),(56,67),(57,67),(58,67),(59,67),(60,67),(61,67),(62,64),(62,66),(63,65),(63,66),(64,67),(65,67),(66,67)],68)
=> ? = 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,15),(1,26),(1,27),(1,32),(1,33),(1,41),(1,42),(1,91),(1,94),(2,14),(2,23),(2,25),(2,29),(2,31),(2,40),(2,42),(2,90),(2,93),(3,13),(3,22),(3,24),(3,28),(3,30),(3,40),(3,41),(3,89),(3,92),(4,18),(4,24),(4,25),(4,34),(4,35),(4,43),(4,44),(4,86),(4,94),(5,17),(5,22),(5,26),(5,36),(5,38),(5,43),(5,45),(5,87),(5,93),(6,16),(6,23),(6,27),(6,37),(6,39),(6,44),(6,45),(6,88),(6,92),(7,21),(7,30),(7,31),(7,36),(7,37),(7,46),(7,47),(7,86),(7,91),(8,20),(8,28),(8,32),(8,34),(8,39),(8,46),(8,48),(8,87),(8,90),(9,19),(9,29),(9,33),(9,35),(9,38),(9,47),(9,48),(9,88),(9,89),(10,19),(10,20),(10,21),(10,49),(10,92),(10,93),(10,94),(11,16),(11,17),(11,18),(11,49),(11,89),(11,90),(11,91),(12,13),(12,14),(12,15),(12,49),(12,86),(12,87),(12,88),(13,53),(13,54),(13,98),(13,100),(13,134),(14,53),(14,55),(14,99),(14,101),(14,135),(15,54),(15,55),(15,102),(15,103),(15,136),(16,56),(16,58),(16,107),(16,108),(16,134),(17,57),(17,58),(17,106),(17,109),(17,135),(18,56),(18,57),(18,104),(18,105),(18,136),(19,59),(19,61),(19,113),(19,114),(19,134),(20,60),(20,61),(20,112),(20,115),(20,135),(21,59),(21,60),(21,110),(21,111),(21,136),(22,63),(22,65),(22,80),(22,98),(22,106),(22,131),(23,64),(23,66),(23,81),(23,99),(23,107),(23,131),(24,62),(24,65),(24,82),(24,100),(24,104),(24,132),(25,62),(25,66),(25,83),(25,101),(25,105),(25,133),(26,63),(26,67),(26,84),(26,102),(26,109),(26,133),(27,64),(27,67),(27,85),(27,103),(27,108),(27,132),(28,69),(28,74),(28,82),(28,98),(28,112),(28,128),(29,70),(29,75),(29,83),(29,99),(29,113),(29,128),(30,68),(30,74),(30,80),(30,100),(30,110),(30,129),(31,68),(31,75),(31,81),(31,101),(31,111),(31,130),(32,69),(32,76),(32,85),(32,102),(32,115),(32,130),(33,70),(33,76),(33,84),(33,103),(33,114),(33,129),(34,72),(34,79),(34,82),(34,105),(34,115),(34,126),(35,71),(35,79),(35,83),(35,104),(35,114),(35,125),(36,73),(36,77),(36,80),(36,109),(36,111),(36,126),(37,73),(37,78),(37,81),(37,108),(37,110),(37,125),(38,71),(38,77),(38,84),(38,106),(38,113),(38,127),(39,72),(39,78),(39,85),(39,107),(39,112),(39,127),(40,50),(40,53),(40,62),(40,68),(40,128),(40,131),(41,50),(41,54),(41,63),(41,69),(41,129),(41,132),(42,50),(42,55),(42,64),(42,70),(42,130),(42,133),(43,51),(43,57),(43,65),(43,71),(43,126),(43,133),(44,51),(44,56),(44,66),(44,72),(44,125),(44,132),(45,51),(45,58),(45,67),(45,73),(45,127),(45,131),(46,52),(46,60),(46,74),(46,78),(46,126),(46,130),(47,52),(47,59),(47,75),(47,77),(47,125),(47,129),(48,52),(48,61),(48,76),(48,79),(48,127),(48,128),(49,134),(49,135),(49,136),(50,95),(50,147),(50,148),(51,96),(51,146),(51,148),(52,97),(52,146),(52,147),(53,95),(53,118),(53,151),(54,95),(54,116),(54,149),(55,95),(55,117),(55,150),(56,96),(56,120),(56,149),(57,96),(57,119),(57,150),(58,96),(58,121),(58,151),(59,97),(59,123),(59,149),(60,97),(60,122),(60,150),(61,97),(61,124),(61,151),(62,118),(62,142),(62,148),(63,116),(63,140),(63,148),(64,117),(64,141),(64,148),(65,119),(65,137),(65,148),(66,120),(66,138),(66,148),(67,121),(67,139),(67,148),(68,118),(68,143),(68,147),(69,116),(69,144),(69,147),(70,117),(70,145),(70,147),(71,119),(71,145),(71,146),(72,120),(72,144),(72,146),(73,121),(73,143),(73,146),(74,122),(74,137),(74,147),(75,123),(75,138),(75,147),(76,124),(76,139),(76,147),(77,123),(77,140),(77,146),(78,122),(78,141),(78,146),(79,124),(79,142),(79,146),(80,137),(80,140),(80,143),(81,138),(81,141),(81,143),(82,137),(82,142),(82,144),(83,138),(83,142),(83,145),(84,139),(84,140),(84,145),(85,139),(85,141),(85,144),(86,100),(86,101),(86,125),(86,126),(86,136),(87,98),(87,102),(87,126),(87,127),(87,135),(88,99),(88,103),(88,125),(88,127),(88,134),(89,104),(89,106),(89,128),(89,129),(89,134),(90,105),(90,107),(90,128),(90,130),(90,135),(91,108),(91,109),(91,129),(91,130),(91,136),(92,110),(92,112),(92,131),(92,132),(92,134),(93,111),(93,113),(93,131),(93,133),(93,135),(94,114),(94,115),(94,132),(94,133),(94,136),(95,152),(96,152),(97,152),(98,116),(98,137),(98,151),(99,117),(99,138),(99,151),(100,118),(100,137),(100,149),(101,118),(101,138),(101,150),(102,116),(102,139),(102,150),(103,117),(103,139),(103,149),(104,119),(104,142),(104,149),(105,120),(105,142),(105,150),(106,119),(106,140),(106,151),(107,120),(107,141),(107,151),(108,121),(108,141),(108,149),(109,121),(109,140),(109,150),(110,122),(110,143),(110,149),(111,123),(111,143),(111,150),(112,122),(112,144),(112,151),(113,123),(113,145),(113,151),(114,124),(114,145),(114,149),(115,124),(115,144),(115,150),(116,152),(117,152),(118,152),(119,152),(120,152),(121,152),(122,152),(123,152),(124,152),(125,138),(125,146),(125,149),(126,137),(126,146),(126,150),(127,139),(127,146),(127,151),(128,142),(128,147),(128,151),(129,140),(129,147),(129,149),(130,141),(130,147),(130,150),(131,143),(131,148),(131,151),(132,144),(132,148),(132,149),(133,145),(133,148),(133,150),(134,149),(134,151),(135,150),(135,151),(136,149),(136,150),(137,152),(138,152),(139,152),(140,152),(141,152),(142,152),(143,152),(144,152),(145,152),(146,152),(147,152),(148,152),(149,152),(150,152),(151,152)],153)
=> ? = 0
([(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,15),(1,20),(1,21),(1,23),(1,29),(1,69),(2,10),(2,14),(2,18),(2,19),(2,22),(2,28),(2,69),(3,13),(3,17),(3,19),(3,21),(3,25),(3,31),(3,68),(4,12),(4,16),(4,18),(4,20),(4,24),(4,30),(4,68),(5,14),(5,15),(5,16),(5,17),(5,27),(5,33),(5,67),(6,10),(6,11),(6,12),(6,13),(6,26),(6,32),(6,67),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,66),(8,22),(8,23),(8,24),(8,25),(8,26),(8,27),(8,66),(9,66),(9,67),(9,68),(9,69),(10,34),(10,46),(10,58),(10,59),(10,87),(11,35),(11,47),(11,60),(11,61),(11,87),(12,36),(12,48),(12,58),(12,60),(12,88),(13,37),(13,49),(13,59),(13,61),(13,88),(14,38),(14,50),(14,62),(14,63),(14,87),(15,39),(15,51),(15,64),(15,65),(15,87),(16,40),(16,52),(16,62),(16,64),(16,88),(17,41),(17,53),(17,63),(17,65),(17,88),(18,42),(18,54),(18,58),(18,62),(18,86),(19,43),(19,55),(19,59),(19,63),(19,86),(20,44),(20,56),(20,60),(20,64),(20,86),(21,45),(21,57),(21,61),(21,65),(21,86),(22,34),(22,38),(22,42),(22,43),(22,89),(23,35),(23,39),(23,44),(23,45),(23,89),(24,36),(24,40),(24,42),(24,44),(24,90),(25,37),(25,41),(25,43),(25,45),(25,90),(26,34),(26,35),(26,36),(26,37),(26,91),(27,38),(27,39),(27,40),(27,41),(27,91),(28,46),(28,50),(28,54),(28,55),(28,89),(29,47),(29,51),(29,56),(29,57),(29,89),(30,48),(30,52),(30,54),(30,56),(30,90),(31,49),(31,53),(31,55),(31,57),(31,90),(32,46),(32,47),(32,48),(32,49),(32,91),(33,50),(33,51),(33,52),(33,53),(33,91),(34,70),(34,71),(34,93),(35,72),(35,73),(35,93),(36,70),(36,72),(36,94),(37,71),(37,73),(37,94),(38,74),(38,75),(38,93),(39,76),(39,77),(39,93),(40,74),(40,76),(40,94),(41,75),(41,77),(41,94),(42,70),(42,74),(42,95),(43,71),(43,75),(43,95),(44,72),(44,76),(44,95),(45,73),(45,77),(45,95),(46,78),(46,79),(46,93),(47,80),(47,81),(47,93),(48,78),(48,80),(48,94),(49,79),(49,81),(49,94),(50,82),(50,83),(50,93),(51,84),(51,85),(51,93),(52,82),(52,84),(52,94),(53,83),(53,85),(53,94),(54,78),(54,82),(54,95),(55,79),(55,83),(55,95),(56,80),(56,84),(56,95),(57,81),(57,85),(57,95),(58,70),(58,78),(58,92),(59,71),(59,79),(59,92),(60,72),(60,80),(60,92),(61,73),(61,81),(61,92),(62,74),(62,82),(62,92),(63,75),(63,83),(63,92),(64,76),(64,84),(64,92),(65,77),(65,85),(65,92),(66,89),(66,90),(66,91),(67,87),(67,88),(67,91),(68,86),(68,88),(68,90),(69,86),(69,87),(69,89),(70,96),(71,96),(72,96),(73,96),(74,96),(75,96),(76,96),(77,96),(78,96),(79,96),(80,96),(81,96),(82,96),(83,96),(84,96),(85,96),(86,92),(86,95),(87,92),(87,93),(88,92),(88,94),(89,93),(89,95),(90,94),(90,95),(91,93),(91,94),(92,96),(93,96),(94,96),(95,96)],97)
=> ? = 0
([(1,5),(2,5),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 0
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,21),(1,25),(1,26),(1,27),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(2,21),(2,22),(2,23),(2,24),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(3,14),(3,19),(3,20),(3,30),(3,36),(3,42),(3,78),(3,79),(4,13),(4,16),(4,18),(4,29),(4,35),(4,41),(4,77),(4,79),(5,12),(5,15),(5,17),(5,28),(5,34),(5,40),(5,77),(5,78),(6,12),(6,18),(6,19),(6,31),(6,37),(6,43),(6,80),(6,82),(7,13),(7,17),(7,20),(7,32),(7,38),(7,44),(7,81),(7,82),(8,14),(8,15),(8,16),(8,33),(8,39),(8,45),(8,80),(8,81),(9,22),(9,25),(9,28),(9,31),(9,46),(9,79),(9,81),(10,23),(10,26),(10,29),(10,32),(10,46),(10,78),(10,80),(11,24),(11,27),(11,30),(11,33),(11,46),(11,77),(11,82),(12,50),(12,116),(12,117),(12,119),(13,51),(13,116),(13,118),(13,120),(14,52),(14,117),(14,118),(14,121),(15,65),(15,71),(15,85),(15,88),(15,117),(16,66),(16,72),(16,86),(16,88),(16,118),(17,67),(17,73),(17,85),(17,89),(17,116),(18,68),(18,74),(18,86),(18,90),(18,116),(19,69),(19,75),(19,87),(19,90),(19,117),(20,70),(20,76),(20,87),(20,89),(20,118),(21,47),(21,48),(21,49),(21,119),(21,120),(21,121),(22,47),(22,53),(22,56),(22,83),(22,93),(22,95),(23,48),(23,54),(23,57),(23,83),(23,92),(23,94),(24,49),(24,55),(24,58),(24,83),(24,91),(24,96),(25,47),(25,59),(25,62),(25,84),(25,99),(25,101),(26,48),(26,60),(26,63),(26,84),(26,98),(26,100),(27,49),(27,61),(27,64),(27,84),(27,97),(27,102),(28,50),(28,53),(28,59),(28,85),(28,126),(29,51),(29,54),(29,60),(29,86),(29,126),(30,52),(30,55),(30,61),(30,87),(30,126),(31,50),(31,56),(31,62),(31,90),(31,125),(32,51),(32,57),(32,63),(32,89),(32,125),(33,52),(33,58),(33,64),(33,88),(33,125),(34,53),(34,65),(34,67),(34,91),(34,92),(34,119),(35,54),(35,66),(35,68),(35,91),(35,93),(35,120),(36,55),(36,69),(36,70),(36,92),(36,93),(36,121),(37,56),(37,68),(37,69),(37,94),(37,96),(37,119),(38,57),(38,67),(38,70),(38,95),(38,96),(38,120),(39,58),(39,65),(39,66),(39,94),(39,95),(39,121),(40,59),(40,71),(40,73),(40,97),(40,98),(40,119),(41,60),(41,72),(41,74),(41,97),(41,99),(41,120),(42,61),(42,75),(42,76),(42,98),(42,99),(42,121),(43,62),(43,74),(43,75),(43,100),(43,102),(43,119),(44,63),(44,73),(44,76),(44,101),(44,102),(44,120),(45,64),(45,71),(45,72),(45,100),(45,101),(45,121),(46,83),(46,84),(46,125),(46,126),(47,103),(47,122),(47,132),(48,103),(48,123),(48,131),(49,103),(49,124),(49,133),(50,122),(50,134),(51,123),(51,134),(52,124),(52,134),(53,104),(53,122),(53,127),(54,105),(54,123),(54,127),(55,106),(55,124),(55,127),(56,109),(56,122),(56,128),(57,108),(57,123),(57,128),(58,107),(58,124),(58,128),(59,110),(59,122),(59,129),(60,111),(60,123),(60,129),(61,112),(61,124),(61,129),(62,115),(62,122),(62,130),(63,114),(63,123),(63,130),(64,113),(64,124),(64,130),(65,104),(65,107),(65,131),(66,105),(66,107),(66,132),(67,104),(67,108),(67,133),(68,105),(68,109),(68,133),(69,106),(69,109),(69,131),(70,106),(70,108),(70,132),(71,110),(71,113),(71,131),(72,111),(72,113),(72,132),(73,110),(73,114),(73,133),(74,111),(74,115),(74,133),(75,112),(75,115),(75,131),(76,112),(76,114),(76,132),(77,88),(77,91),(77,97),(77,116),(77,126),(78,89),(78,92),(78,98),(78,117),(78,126),(79,90),(79,93),(79,99),(79,118),(79,126),(80,86),(80,94),(80,100),(80,117),(80,125),(81,85),(81,95),(81,101),(81,118),(81,125),(82,87),(82,96),(82,102),(82,116),(82,125),(83,103),(83,127),(83,128),(84,103),(84,129),(84,130),(85,104),(85,110),(85,134),(86,105),(86,111),(86,134),(87,106),(87,112),(87,134),(88,107),(88,113),(88,134),(89,108),(89,114),(89,134),(90,109),(90,115),(90,134),(91,107),(91,127),(91,133),(92,108),(92,127),(92,131),(93,109),(93,127),(93,132),(94,105),(94,128),(94,131),(95,104),(95,128),(95,132),(96,106),(96,128),(96,133),(97,113),(97,129),(97,133),(98,114),(98,129),(98,131),(99,115),(99,129),(99,132),(100,111),(100,130),(100,131),(101,110),(101,130),(101,132),(102,112),(102,130),(102,133),(103,135),(104,135),(105,135),(106,135),(107,135),(108,135),(109,135),(110,135),(111,135),(112,135),(113,135),(114,135),(115,135),(116,133),(116,134),(117,131),(117,134),(118,132),(118,134),(119,122),(119,131),(119,133),(120,123),(120,132),(120,133),(121,124),(121,131),(121,132),(122,135),(123,135),(124,135),(125,128),(125,130),(125,134),(126,127),(126,129),(126,134),(127,135),(128,135),(129,135),(130,135),(131,135),(132,135),(133,135),(134,135)],136)
=> ? = 0
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 0
([(1,5),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,22),(1,27),(1,28),(1,32),(1,35),(1,36),(1,37),(1,40),(1,41),(1,90),(2,21),(2,25),(2,26),(2,31),(2,33),(2,34),(2,37),(2,38),(2,39),(2,89),(3,16),(3,26),(3,30),(3,41),(3,43),(3,45),(3,49),(3,86),(3,92),(4,15),(4,25),(4,29),(4,40),(4,42),(4,44),(4,48),(4,85),(4,92),(5,17),(5,27),(5,29),(5,39),(5,43),(5,46),(5,50),(5,88),(5,91),(6,18),(6,28),(6,30),(6,38),(6,42),(6,47),(6,51),(6,87),(6,91),(7,13),(7,23),(7,24),(7,44),(7,45),(7,46),(7,47),(7,89),(7,90),(8,13),(8,19),(8,20),(8,21),(8,22),(8,59),(8,91),(8,92),(9,14),(9,17),(9,18),(9,19),(9,31),(9,85),(9,86),(9,90),(10,14),(10,15),(10,16),(10,20),(10,32),(10,87),(10,88),(10,89),(11,24),(11,33),(11,35),(11,48),(11,50),(11,59),(11,86),(11,87),(12,23),(12,34),(12,36),(12,49),(12,51),(12,59),(12,85),(12,88),(13,97),(13,98),(13,99),(13,100),(13,103),(14,52),(14,127),(14,128),(14,138),(15,77),(15,96),(15,106),(15,112),(15,127),(16,78),(16,96),(16,107),(16,113),(16,128),(17,75),(17,95),(17,108),(17,111),(17,127),(18,76),(18,95),(18,109),(18,110),(18,128),(19,52),(19,69),(19,95),(19,100),(19,139),(20,52),(20,70),(20,96),(20,99),(20,140),(21,68),(21,69),(21,99),(21,101),(21,104),(21,114),(22,68),(22,70),(22,100),(22,102),(22,105),(22,115),(23,65),(23,67),(23,103),(23,134),(23,136),(24,64),(24,66),(24,103),(24,135),(24,137),(25,55),(25,71),(25,104),(25,106),(25,116),(25,132),(26,56),(26,72),(26,104),(26,107),(26,117),(26,133),(27,58),(27,73),(27,105),(27,108),(27,119),(27,132),(28,57),(28,74),(28,105),(28,109),(28,118),(28,133),(29,60),(29,62),(29,127),(29,129),(29,132),(30,61),(30,63),(30,128),(30,129),(30,133),(31,69),(31,75),(31,76),(31,116),(31,117),(31,138),(32,70),(32,77),(32,78),(32,118),(32,119),(32,138),(33,71),(33,80),(33,84),(33,114),(33,117),(33,135),(34,72),(34,79),(34,83),(34,114),(34,116),(34,134),(35,73),(35,81),(35,84),(35,115),(35,118),(35,137),(36,74),(36,82),(36,83),(36,115),(36,119),(36,136),(37,68),(37,83),(37,84),(37,132),(37,133),(37,138),(38,55),(38,76),(38,79),(38,101),(38,133),(38,135),(39,56),(39,75),(39,80),(39,101),(39,132),(39,134),(40,57),(40,77),(40,81),(40,102),(40,132),(40,136),(41,58),(41,78),(41,82),(41,102),(41,133),(41,137),(42,53),(42,55),(42,57),(42,110),(42,112),(42,129),(43,54),(43,56),(43,58),(43,111),(43,113),(43,129),(44,53),(44,62),(44,66),(44,97),(44,106),(44,136),(45,54),(45,63),(45,67),(45,97),(45,107),(45,137),(46,54),(46,62),(46,64),(46,98),(46,108),(46,134),(47,53),(47,63),(47,65),(47,98),(47,109),(47,135),(48,60),(48,66),(48,71),(48,81),(48,112),(48,139),(49,61),(49,67),(49,72),(49,82),(49,113),(49,139),(50,60),(50,64),(50,73),(50,80),(50,111),(50,140),(51,61),(51,65),(51,74),(51,79),(51,110),(51,140),(52,144),(52,156),(53,143),(53,145),(53,147),(54,143),(54,146),(54,148),(55,121),(55,145),(55,151),(56,122),(56,146),(56,151),(57,123),(57,147),(57,151),(58,124),(58,148),(58,151),(59,103),(59,114),(59,115),(59,139),(59,140),(60,93),(60,141),(60,156),(61,94),(61,142),(61,156),(62,93),(62,143),(62,154),(63,94),(63,143),(63,155),(64,93),(64,148),(64,152),(65,94),(65,147),(65,152),(66,93),(66,145),(66,153),(67,94),(67,146),(67,153),(68,120),(68,144),(68,151),(69,125),(69,144),(69,149),(70,126),(70,144),(70,150),(71,141),(71,145),(71,149),(72,142),(72,146),(72,149),(73,141),(73,148),(73,150),(74,142),(74,147),(74,150),(75,122),(75,125),(75,154),(76,121),(76,125),(76,155),(77,123),(77,126),(77,154),(78,124),(78,126),(78,155),(79,121),(79,142),(79,152),(80,122),(80,141),(80,152),(81,123),(81,141),(81,153),(82,124),(82,142),(82,153),(83,120),(83,142),(83,154),(84,120),(84,141),(84,155),(85,110),(85,116),(85,127),(85,136),(85,139),(86,111),(86,117),(86,128),(86,137),(86,139),(87,112),(87,118),(87,128),(87,135),(87,140),(88,113),(88,119),(88,127),(88,134),(88,140),(89,99),(89,106),(89,107),(89,134),(89,135),(89,138),(90,100),(90,108),(90,109),(90,136),(90,137),(90,138),(91,95),(91,98),(91,101),(91,105),(91,129),(91,140),(92,96),(92,97),(92,102),(92,104),(92,129),(92,139),(93,157),(94,157),(95,125),(95,131),(95,156),(96,126),(96,130),(96,156),(97,130),(97,143),(97,153),(98,131),(98,143),(98,152),(99,130),(99,144),(99,152),(100,131),(100,144),(100,153),(101,125),(101,151),(101,152),(102,126),(102,151),(102,153),(103,152),(103,153),(104,130),(104,149),(104,151),(105,131),(105,150),(105,151),(106,130),(106,145),(106,154),(107,130),(107,146),(107,155),(108,131),(108,148),(108,154),(109,131),(109,147),(109,155),(110,121),(110,147),(110,156),(111,122),(111,148),(111,156),(112,123),(112,145),(112,156),(113,124),(113,146),(113,156),(114,120),(114,149),(114,152),(115,120),(115,150),(115,153),(116,121),(116,149),(116,154),(117,122),(117,149),(117,155),(118,123),(118,150),(118,155),(119,124),(119,150),(119,154),(120,157),(121,157),(122,157),(123,157),(124,157),(125,157),(126,157),(127,154),(127,156),(128,155),(128,156),(129,143),(129,151),(129,156),(130,157),(131,157),(132,141),(132,151),(132,154),(133,142),(133,151),(133,155),(134,146),(134,152),(134,154),(135,145),(135,152),(135,155),(136,147),(136,153),(136,154),(137,148),(137,153),(137,155),(138,144),(138,154),(138,155),(139,149),(139,153),(139,156),(140,150),(140,152),(140,156),(141,157),(142,157),(143,157),(144,157),(145,157),(146,157),(147,157),(148,157),(149,157),(150,157),(151,157),(152,157),(153,157),(154,157),(155,157),(156,157)],158)
=> ? = 0
([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,13),(1,15),(1,19),(1,23),(1,27),(1,31),(1,38),(1,39),(2,13),(2,14),(2,18),(2,22),(2,26),(2,30),(2,36),(2,37),(3,12),(3,17),(3,21),(3,25),(3,29),(3,33),(3,37),(3,39),(4,12),(4,16),(4,20),(4,24),(4,28),(4,32),(4,36),(4,38),(5,11),(5,22),(5,23),(5,24),(5,25),(5,35),(5,41),(6,11),(6,18),(6,19),(6,20),(6,21),(6,34),(6,40),(7,10),(7,26),(7,27),(7,28),(7,29),(7,34),(7,41),(8,10),(8,30),(8,31),(8,32),(8,33),(8,35),(8,40),(9,14),(9,15),(9,16),(9,17),(9,40),(9,41),(10,96),(10,100),(10,101),(11,96),(11,98),(11,99),(12,43),(12,97),(12,99),(12,101),(13,42),(13,97),(13,98),(13,100),(14,42),(14,44),(14,45),(14,80),(14,84),(15,42),(15,46),(15,47),(15,81),(15,85),(16,43),(16,44),(16,46),(16,82),(16,86),(17,43),(17,45),(17,47),(17,83),(17,87),(18,48),(18,49),(18,64),(18,80),(18,98),(19,50),(19,51),(19,65),(19,81),(19,98),(20,48),(20,50),(20,66),(20,82),(20,99),(21,49),(21,51),(21,67),(21,83),(21,99),(22,52),(22,53),(22,68),(22,84),(22,98),(23,54),(23,55),(23,69),(23,85),(23,98),(24,52),(24,54),(24,70),(24,86),(24,99),(25,53),(25,55),(25,71),(25,87),(25,99),(26,56),(26,57),(26,64),(26,84),(26,100),(27,58),(27,59),(27,65),(27,85),(27,100),(28,56),(28,58),(28,66),(28,86),(28,101),(29,57),(29,59),(29,67),(29,87),(29,101),(30,60),(30,61),(30,68),(30,80),(30,100),(31,62),(31,63),(31,69),(31,81),(31,100),(32,60),(32,62),(32,70),(32,82),(32,101),(33,61),(33,63),(33,71),(33,83),(33,101),(34,64),(34,65),(34,66),(34,67),(34,96),(35,68),(35,69),(35,70),(35,71),(35,96),(36,44),(36,48),(36,52),(36,56),(36,60),(36,97),(37,45),(37,49),(37,53),(37,57),(37,61),(37,97),(38,46),(38,50),(38,54),(38,58),(38,62),(38,97),(39,47),(39,51),(39,55),(39,59),(39,63),(39,97),(40,80),(40,81),(40,82),(40,83),(40,96),(41,84),(41,85),(41,86),(41,87),(41,96),(42,102),(42,105),(43,102),(43,106),(44,88),(44,92),(44,102),(45,89),(45,93),(45,102),(46,90),(46,94),(46,102),(47,91),(47,95),(47,102),(48,72),(48,88),(48,103),(49,73),(49,89),(49,103),(50,74),(50,90),(50,103),(51,75),(51,91),(51,103),(52,76),(52,92),(52,103),(53,77),(53,93),(53,103),(54,78),(54,94),(54,103),(55,79),(55,95),(55,103),(56,72),(56,92),(56,104),(57,73),(57,93),(57,104),(58,74),(58,94),(58,104),(59,75),(59,95),(59,104),(60,76),(60,88),(60,104),(61,77),(61,89),(61,104),(62,78),(62,90),(62,104),(63,79),(63,91),(63,104),(64,72),(64,73),(64,105),(65,74),(65,75),(65,105),(66,72),(66,74),(66,106),(67,73),(67,75),(67,106),(68,76),(68,77),(68,105),(69,78),(69,79),(69,105),(70,76),(70,78),(70,106),(71,77),(71,79),(71,106),(72,107),(73,107),(74,107),(75,107),(76,107),(77,107),(78,107),(79,107),(80,88),(80,89),(80,105),(81,90),(81,91),(81,105),(82,88),(82,90),(82,106),(83,89),(83,91),(83,106),(84,92),(84,93),(84,105),(85,94),(85,95),(85,105),(86,92),(86,94),(86,106),(87,93),(87,95),(87,106),(88,107),(89,107),(90,107),(91,107),(92,107),(93,107),(94,107),(95,107),(96,105),(96,106),(97,102),(97,103),(97,104),(98,103),(98,105),(99,103),(99,106),(100,104),(100,105),(101,104),(101,106),(102,107),(103,107),(104,107),(105,107),(106,107)],108)
=> ? = 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,20),(1,21),(1,32),(1,35),(1,36),(1,41),(1,42),(1,43),(2,15),(2,18),(2,19),(2,31),(2,33),(2,34),(2,39),(2,40),(2,43),(3,23),(3,25),(3,28),(3,38),(3,40),(3,42),(3,71),(3,75),(4,22),(4,24),(4,27),(4,37),(4,39),(4,41),(4,70),(4,75),(5,17),(5,22),(5,23),(5,26),(5,31),(5,32),(5,72),(5,73),(6,13),(6,24),(6,29),(6,33),(6,35),(6,38),(6,72),(6,74),(7,12),(7,25),(7,30),(7,34),(7,36),(7,37),(7,73),(7,74),(8,12),(8,13),(8,14),(8,17),(8,43),(8,70),(8,71),(9,14),(9,15),(9,16),(9,26),(9,44),(9,74),(9,75),(10,18),(10,20),(10,27),(10,30),(10,44),(10,71),(10,72),(11,19),(11,21),(11,28),(11,29),(11,44),(11,70),(11,73),(12,80),(12,88),(12,94),(12,108),(13,80),(13,87),(13,95),(13,109),(14,45),(14,80),(14,83),(14,117),(15,62),(15,81),(15,83),(15,84),(15,92),(16,63),(16,82),(16,83),(16,85),(16,93),(17,45),(17,91),(17,108),(17,109),(18,58),(18,65),(18,84),(18,96),(18,111),(19,59),(19,64),(19,84),(19,97),(19,110),(20,60),(20,67),(20,85),(20,99),(20,111),(21,61),(21,66),(21,85),(21,98),(21,110),(22,46),(22,48),(22,86),(22,89),(22,108),(23,47),(23,49),(23,86),(23,90),(23,109),(24,50),(24,52),(24,87),(24,89),(24,107),(25,51),(25,53),(25,88),(25,90),(25,107),(26,45),(26,62),(26,63),(26,86),(26,118),(27,58),(27,60),(27,68),(27,89),(27,117),(28,59),(28,61),(28,69),(28,90),(28,117),(29,64),(29,66),(29,69),(29,87),(29,118),(30,65),(30,67),(30,68),(30,88),(30,118),(31,46),(31,47),(31,62),(31,91),(31,96),(31,97),(32,48),(32,49),(32,63),(32,91),(32,98),(32,99),(33,50),(33,55),(33,64),(33,92),(33,95),(33,96),(34,51),(34,54),(34,65),(34,92),(34,94),(34,97),(35,52),(35,57),(35,66),(35,93),(35,95),(35,99),(36,53),(36,56),(36,67),(36,93),(36,94),(36,98),(37,54),(37,56),(37,68),(37,107),(37,108),(38,55),(38,57),(38,69),(38,107),(38,109),(39,46),(39,50),(39,54),(39,58),(39,81),(39,110),(40,47),(40,51),(40,55),(40,59),(40,81),(40,111),(41,48),(41,52),(41,56),(41,60),(41,82),(41,110),(42,49),(42,53),(42,57),(42,61),(42,82),(42,111),(43,83),(43,91),(43,94),(43,95),(43,110),(43,111),(44,84),(44,85),(44,117),(44,118),(45,106),(45,124),(46,100),(46,104),(46,121),(47,101),(47,104),(47,122),(48,102),(48,105),(48,121),(49,103),(49,105),(49,122),(50,100),(50,113),(50,115),(51,101),(51,114),(51,115),(52,102),(52,113),(52,116),(53,103),(53,114),(53,116),(54,76),(54,115),(54,121),(55,77),(55,115),(55,122),(56,78),(56,116),(56,121),(57,79),(57,116),(57,122),(58,76),(58,100),(58,123),(59,77),(59,101),(59,123),(60,78),(60,102),(60,123),(61,79),(61,103),(61,123),(62,104),(62,106),(62,119),(63,105),(63,106),(63,120),(64,77),(64,113),(64,119),(65,76),(65,114),(65,119),(66,79),(66,113),(66,120),(67,78),(67,114),(67,120),(68,76),(68,78),(68,124),(69,77),(69,79),(69,124),(70,87),(70,108),(70,110),(70,117),(71,88),(71,109),(71,111),(71,117),(72,89),(72,96),(72,99),(72,109),(72,118),(73,90),(73,97),(73,98),(73,108),(73,118),(74,80),(74,92),(74,93),(74,107),(74,118),(75,81),(75,82),(75,86),(75,107),(75,117),(76,125),(77,125),(78,125),(79,125),(80,112),(80,124),(81,104),(81,115),(81,123),(82,105),(82,116),(82,123),(83,106),(83,112),(83,123),(84,119),(84,123),(85,120),(85,123),(86,104),(86,105),(86,124),(87,113),(87,124),(88,114),(88,124),(89,100),(89,102),(89,124),(90,101),(90,103),(90,124),(91,106),(91,121),(91,122),(92,112),(92,115),(92,119),(93,112),(93,116),(93,120),(94,112),(94,114),(94,121),(95,112),(95,113),(95,122),(96,100),(96,119),(96,122),(97,101),(97,119),(97,121),(98,103),(98,120),(98,121),(99,102),(99,120),(99,122),(100,125),(101,125),(102,125),(103,125),(104,125),(105,125),(106,125),(107,115),(107,116),(107,124),(108,121),(108,124),(109,122),(109,124),(110,113),(110,121),(110,123),(111,114),(111,122),(111,123),(112,125),(113,125),(114,125),(115,125),(116,125),(117,123),(117,124),(118,119),(118,120),(118,124),(119,125),(120,125),(121,125),(122,125),(123,125),(124,125)],126)
=> ? = 0
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(1,9),(1,36),(1,37),(2,12),(2,16),(2,21),(2,22),(2,37),(3,11),(3,15),(3,19),(3,20),(3,37),(4,14),(4,18),(4,20),(4,22),(4,36),(5,13),(5,17),(5,19),(5,21),(5,36),(6,9),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(8,35),(8,38),(8,39),(9,35),(9,40),(9,41),(10,31),(10,32),(10,33),(10,34),(10,35),(11,23),(11,24),(11,31),(11,38),(12,25),(12,26),(12,32),(12,38),(13,23),(13,25),(13,33),(13,39),(14,24),(14,26),(14,34),(14,39),(15,27),(15,28),(15,31),(15,40),(16,29),(16,30),(16,32),(16,40),(17,27),(17,29),(17,33),(17,41),(18,28),(18,30),(18,34),(18,41),(19,23),(19,27),(19,48),(20,24),(20,28),(20,48),(21,25),(21,29),(21,48),(22,26),(22,30),(22,48),(23,42),(23,49),(24,43),(24,49),(25,44),(25,49),(26,45),(26,49),(27,42),(27,50),(28,43),(28,50),(29,44),(29,50),(30,45),(30,50),(31,42),(31,43),(31,46),(32,44),(32,45),(32,46),(33,42),(33,44),(33,47),(34,43),(34,45),(34,47),(35,46),(35,47),(36,39),(36,41),(36,48),(37,38),(37,40),(37,48),(38,46),(38,49),(39,47),(39,49),(40,46),(40,50),(41,47),(41,50),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,49),(48,50),(49,51),(50,51)],52)
=> ? = 0
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,14),(1,19),(1,20),(1,21),(1,22),(1,35),(1,36),(2,10),(2,14),(2,15),(2,16),(2,17),(2,18),(2,33),(2,34),(3,13),(3,16),(3,20),(3,24),(3,26),(3,30),(3,70),(4,13),(4,15),(4,19),(4,23),(4,25),(4,29),(4,69),(5,12),(5,17),(5,21),(5,23),(5,27),(5,31),(5,70),(6,12),(6,18),(6,22),(6,24),(6,28),(6,32),(6,69),(7,29),(7,30),(7,31),(7,32),(7,34),(7,36),(7,68),(8,25),(8,26),(8,27),(8,28),(8,33),(8,35),(8,68),(9,10),(9,11),(9,68),(9,69),(9,70),(10,37),(10,79),(10,80),(10,83),(11,37),(11,81),(11,82),(11,84),(12,61),(12,63),(12,87),(12,88),(13,60),(13,62),(13,86),(13,88),(14,37),(14,38),(14,39),(14,86),(14,87),(15,40),(15,44),(15,56),(15,79),(15,86),(16,41),(16,45),(16,57),(16,80),(16,86),(17,42),(17,47),(17,56),(17,80),(17,87),(18,43),(18,46),(18,57),(18,79),(18,87),(19,48),(19,52),(19,58),(19,81),(19,86),(20,49),(20,53),(20,59),(20,82),(20,86),(21,50),(21,55),(21,58),(21,82),(21,87),(22,51),(22,54),(22,59),(22,81),(22,87),(23,56),(23,58),(23,64),(23,66),(23,88),(24,57),(24,59),(24,65),(24,67),(24,88),(25,40),(25,48),(25,60),(25,64),(25,89),(26,41),(26,49),(26,60),(26,65),(26,90),(27,42),(27,50),(27,61),(27,64),(27,90),(28,43),(28,51),(28,61),(28,65),(28,89),(29,44),(29,52),(29,62),(29,66),(29,89),(30,45),(30,53),(30,62),(30,67),(30,90),(31,47),(31,55),(31,63),(31,66),(31,90),(32,46),(32,54),(32,63),(32,67),(32,89),(33,38),(33,40),(33,41),(33,42),(33,43),(33,83),(34,39),(34,44),(34,45),(34,46),(34,47),(34,83),(35,38),(35,48),(35,49),(35,50),(35,51),(35,84),(36,39),(36,52),(36,53),(36,54),(36,55),(36,84),(37,85),(37,100),(38,85),(38,91),(38,92),(39,85),(39,93),(39,94),(40,71),(40,91),(40,95),(41,72),(41,91),(41,96),(42,71),(42,92),(42,96),(43,72),(43,92),(43,95),(44,73),(44,93),(44,95),(45,74),(45,93),(45,96),(46,74),(46,94),(46,95),(47,73),(47,94),(47,96),(48,75),(48,91),(48,97),(49,76),(49,91),(49,98),(50,75),(50,92),(50,98),(51,76),(51,92),(51,97),(52,77),(52,93),(52,97),(53,78),(53,93),(53,98),(54,78),(54,94),(54,97),(55,77),(55,94),(55,98),(56,71),(56,73),(56,100),(57,72),(57,74),(57,100),(58,75),(58,77),(58,100),(59,76),(59,78),(59,100),(60,91),(60,99),(61,92),(61,99),(62,93),(62,99),(63,94),(63,99),(64,71),(64,75),(64,99),(65,72),(65,76),(65,99),(66,73),(66,77),(66,99),(67,74),(67,78),(67,99),(68,83),(68,84),(68,89),(68,90),(69,79),(69,81),(69,88),(69,89),(70,80),(70,82),(70,88),(70,90),(71,101),(72,101),(73,101),(74,101),(75,101),(76,101),(77,101),(78,101),(79,95),(79,100),(80,96),(80,100),(81,97),(81,100),(82,98),(82,100),(83,85),(83,95),(83,96),(84,85),(84,97),(84,98),(85,101),(86,91),(86,93),(86,100),(87,92),(87,94),(87,100),(88,99),(88,100),(89,95),(89,97),(89,99),(90,96),(90,98),(90,99),(91,101),(92,101),(93,101),(94,101),(95,101),(96,101),(97,101),(98,101),(99,101),(100,101)],102)
=> ? = 0
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
Matching statistic: St000379
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(1,5),(2,5),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,5),(2,5),(3,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ? = 2
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,5),(2,5),(3,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001570
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 1
([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,5),(2,5),(3,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1.
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