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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000071
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Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([],3)
=> ([(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
([(1,2)],3)
=> ([(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)
=> 3
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(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
([(0,2),(0,3),(3,1)],4)
=> ([(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
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(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)
=> 2
([(1,2),(2,3)],4)
=> ([(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)
=> 4
([(0,3),(3,1),(3,2)],4)
=> ([(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)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(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)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
([(0,3),(1,2)],4)
=> ([(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)
=> 6
([(0,3),(1,2),(1,3)],4)
=> ([(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)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(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)
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(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
([(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)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(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)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(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
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(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
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(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)
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(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)
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(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
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(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)
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(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)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(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
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(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)
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(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)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
([(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)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St000909
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([],3)
=> ([(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
([(1,2)],3)
=> ([(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)
=> 3
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(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
([(0,2),(0,3),(3,1)],4)
=> ([(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
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(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)
=> 2
([(1,2),(2,3)],4)
=> ([(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)
=> 4
([(0,3),(3,1),(3,2)],4)
=> ([(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)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(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)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
([(0,3),(1,2)],4)
=> ([(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)
=> 6
([(0,3),(1,2),(1,3)],4)
=> ([(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)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(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)
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(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
([(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)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(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)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(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
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(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
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(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)
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(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)
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(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
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(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)
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(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)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(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
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(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)
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(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)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
([(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)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000100
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ? = 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 6
([(1,2)],3)
=> 3
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3)],4)
=> 6
([(0,2),(0,3),(3,1)],4)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 4
([(0,3),(3,1),(3,2)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 6
([(0,3),(1,2)],4)
=> 6
([(0,3),(1,2),(1,3)],4)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
Description
The number of linear extensions of a poset.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([],3)
=> [3,3]
=> 6
([(1,2)],3)
=> [3]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 6
([(0,3),(1,2)],4)
=> [4,2]
=> 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? = 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 1
([],2)
=> [2]
=> 100 => 2
([(0,1)],2)
=> [1]
=> 10 => 1
([],3)
=> [3,3]
=> 11000 => 6
([(1,2)],3)
=> [3]
=> 1000 => 3
([(0,1),(0,2)],3)
=> [2]
=> 100 => 2
([(0,2),(2,1)],3)
=> [1]
=> 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 11000 => 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1000 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 100 => 2
([(1,2),(2,3)],4)
=> [4]
=> 10000 => 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 100 => 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 100 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => 6
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 10 => 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 100 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 1100 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 1100 => 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 100 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1000000 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 1100 => 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 1000 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1000000 => 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 100 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1000 => 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1000 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 10 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 10000 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 100 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 100 => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 10 => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 10 => 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? => ? = 1
Description
The number of inversions of a binary word.
Matching statistic: St000531
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [2]
=> [1,1]
=> 2
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> [3,3]
=> [6]
=> 6
([(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [6]
=> 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [6]
=> 6
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [5]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [5]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [4]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [4]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [4]
=> 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [1,1]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> [1]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> [1]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ?
=> ? = 1
Description
The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St001034
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 1
([],2)
=> [2]
=> [1,0,1,0]
=> 2
([(0,1)],2)
=> [1]
=> [1,0]
=> 1
([],3)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
([(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> [1,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
([(0,3),(1,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,0,1,0]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,0,1,0]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1,0]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,0,1,0]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [1,0,1,0]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> [1,0]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> [1,0]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ?
=> ? = 1
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001659
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [2]
=> [1,1]
=> 2
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> [3,3]
=> [6]
=> 6
([(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(0,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [6]
=> 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [6]
=> 6
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [5]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [5]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [4]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [4]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [4]
=> 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [1,1]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> [1]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> [1]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ?
=> ? = 1
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St000290
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 10 => 1
([],2)
=> [2]
=> 100 => 010 => 2
([(0,1)],2)
=> [1]
=> 10 => 10 => 1
([],3)
=> [3,3]
=> 11000 => 01010 => 6
([(1,2)],3)
=> [3]
=> 1000 => 0010 => 3
([(0,1),(0,2)],3)
=> [2]
=> 100 => 010 => 2
([(0,2),(2,1)],3)
=> [1]
=> 10 => 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 010 => 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 11000 => 01010 => 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 100 => 010 => 2
([(1,2),(2,3)],4)
=> [4]
=> 10000 => 00010 => 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => 01010 => 6
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => 10010 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1010 => 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 10 => 10 => 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 100 => 010 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 10 => 10 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 100 => 010 => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 10 => 10 => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 10 => 10 => 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? => ? => ? = 1
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000391
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 10 => 1
([],2)
=> [2]
=> 100 => 010 => 2
([(0,1)],2)
=> [1]
=> 10 => 10 => 1
([],3)
=> [3,3]
=> 11000 => 01010 => 6
([(1,2)],3)
=> [3]
=> 1000 => 0010 => 3
([(0,1),(0,2)],3)
=> [2]
=> 100 => 010 => 2
([(0,2),(2,1)],3)
=> [1]
=> 10 => 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 010 => 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 11000 => 01010 => 6
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 100 => 010 => 2
([(1,2),(2,3)],4)
=> [4]
=> 10000 => 00010 => 4
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => 01010 => 6
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => 10010 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1010 => 4
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 10 => 10 => 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 100 => 010 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 1100 => 1010 => 4
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 10 => 10 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 100 => 010 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 100 => 010 => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 10 => 10 => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 10 => 10 => 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? => ? => ? = 1
Description
The sum of the positions of the ones in a binary word.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000395The sum of the heights of the peaks of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000867The sum of the hook lengths in the first row of an integer partition. St000363The number of minimal vertex covers of a graph. St001304The number of maximally independent sets of vertices of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000450The number of edges minus the number of vertices plus 2 of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001624The breadth of a lattice. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.
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