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Your data matches 76 different statistics following compositions of up to 3 maps.
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Matching statistic: St001625
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Values
[]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> -1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> -1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> -1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(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
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> -1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],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)
=> 0
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,1)],2)
=> -1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001878
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(load all 3 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = -1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? = -1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = -1 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 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
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? = -1 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(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
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,1)],2)
=> ? = -1 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(0,1)],2)
=> ? = -1 + 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,1)],2)
=> ? = -1 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000480
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[]
=> .
=> ? => ?
=> ? = 1 + 1
[[]]
=> [.,.]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 1 = 0 + 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 1 = 0 + 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 1 = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [8,9,7,10,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [9,10,6,7,5,8,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [8,9,6,7,5,10,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [7,8,6,9,5,10,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [8,9,6,7,4,5,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [7,8,6,9,4,5,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [8,9,5,6,4,7,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [7,8,5,6,4,9,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [6,7,5,8,4,9,3,10,1,2] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [8,9,6,7,4,5,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [7,8,6,9,4,5,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [8,9,5,6,4,7,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [7,8,5,6,4,9,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [6,7,5,8,4,9,2,3,1,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
=> [7,8,5,6,3,4,2,9,1,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
=> [6,7,5,8,3,4,2,9,1,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],[[.,.],.]]],.]],.]
=> [7,8,4,5,3,6,2,9,1,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
=> [6,7,4,5,3,8,2,9,1,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]]
=> [10,11,9,12,7,8,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]]
=> [10,11,8,9,7,12,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]]
=> [9,10,8,11,7,12,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]]
=> [10,11,8,9,6,7,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]]
=> [9,10,8,11,6,7,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]]
=> [9,10,7,8,6,11,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]]
=> [8,9,7,10,6,11,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]]
=> [10,11,8,9,6,7,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]]
=> [9,10,8,11,6,7,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]]
=> [9,10,7,8,6,11,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]]
=> [8,9,7,10,6,11,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]]
=> [9,10,7,8,5,6,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]]
=> [8,9,7,10,5,6,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]]
=> [8,9,6,7,5,10,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]]
=> [7,8,6,9,5,10,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]],.]
=> [10,11,8,9,6,7,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]],.]
=> [9,10,8,11,6,7,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]],.]
=> [9,10,7,8,6,11,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]],.]
=> [8,9,7,10,6,11,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]],.]
=> [9,10,7,8,5,6,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]],.]
=> [8,9,7,10,5,6,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]],.]
=> [8,9,6,7,5,10,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]],.]
=> [7,8,6,9,5,10,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[[],[[],[[],[[],[]]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]],.]
=> [9,10,7,8,5,6,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[],[[[],[]],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]],.]
=> [8,9,7,10,5,6,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[[],[[],[]]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]],.]
=> [8,9,6,7,5,10,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[[[],[]],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]],.]
=> [7,8,6,9,5,10,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[],[[],[[],[]]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]],.]
=> [8,9,6,7,4,5,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[],[[[],[]],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]],.]
=> [7,8,6,9,4,5,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[[],[[],[]]],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]],.]
=> [7,8,5,6,4,9,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[[[],[]],[]],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]],.]
=> [6,7,5,8,4,9,3,10,2,11,1,12] => ?
=> ? = 0 + 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St001280
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[]
=> .
=> ? => ?
=> ? = 1 + 1
[[]]
=> [.,.]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 1 = 0 + 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 1 = 0 + 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 1 = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [8,9,7,10,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [9,10,6,7,5,8,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [8,9,6,7,5,10,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [7,8,6,9,5,10,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [8,9,6,7,4,5,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [7,8,6,9,4,5,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [8,9,5,6,4,7,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [7,8,5,6,4,9,3,10,1,2] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [6,7,5,8,4,9,3,10,1,2] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [8,9,6,7,4,5,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [7,8,6,9,4,5,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [8,9,5,6,4,7,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [7,8,5,6,4,9,2,3,1,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [6,7,5,8,4,9,2,3,1,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
=> [7,8,5,6,3,4,2,9,1,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
=> [6,7,5,8,3,4,2,9,1,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],[[.,.],.]]],.]],.]
=> [7,8,4,5,3,6,2,9,1,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
=> [6,7,4,5,3,8,2,9,1,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]]
=> [10,11,9,12,7,8,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]]
=> [10,11,8,9,7,12,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]]
=> [9,10,8,11,7,12,5,6,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]]
=> [10,11,8,9,6,7,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]]
=> [9,10,8,11,6,7,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]]
=> [9,10,7,8,6,11,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]]
=> [8,9,7,10,6,11,5,12,3,4,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]]
=> [10,11,8,9,6,7,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]]
=> [9,10,8,11,6,7,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]]
=> [9,10,7,8,6,11,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]]
=> [8,9,7,10,6,11,4,5,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]]
=> [9,10,7,8,5,6,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]]
=> [8,9,7,10,5,6,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]]
=> [8,9,6,7,5,10,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]]
=> [7,8,6,9,5,10,4,11,3,12,1,2] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]],.]
=> [10,11,8,9,6,7,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]],.]
=> [9,10,8,11,6,7,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]],.]
=> [9,10,7,8,6,11,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]],.]
=> [8,9,7,10,6,11,4,5,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]],.]
=> [9,10,7,8,5,6,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]],.]
=> [8,9,7,10,5,6,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]],.]
=> [8,9,6,7,5,10,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]],.]
=> [7,8,6,9,5,10,4,11,2,3,1,12] => ?
=> ? = 0 + 1
[[[[],[[],[[],[[],[]]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]],.]
=> [9,10,7,8,5,6,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[],[[[],[]],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]],.]
=> [8,9,7,10,5,6,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[[],[[],[]]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]],.]
=> [8,9,6,7,5,10,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[],[[[[],[]],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]],.]
=> [7,8,6,9,5,10,3,4,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[],[[],[[],[]]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]],.]
=> [8,9,6,7,4,5,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[],[[[],[]],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]],.]
=> [7,8,6,9,4,5,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[[],[[],[]]],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]],.]
=> [7,8,5,6,4,9,3,10,2,11,1,12] => ?
=> ? = 0 + 1
[[[[[[[],[]],[]],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]],.]
=> [6,7,5,8,4,9,3,10,2,11,1,12] => ?
=> ? = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000481
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
[]
=> .
=> ? => ?
=> ? = 1 + 1
[[]]
=> [.,.]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 0 = -1 + 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 0 = -1 + 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 0 = -1 + 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 0 + 1
[[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ?
=> ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 0 + 1
[[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [7,5,4,6,8,3,9,2,10,1] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [7,6,8,4,3,5,9,2,10,1] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [8,5,4,6,3,7,9,2,10,1] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [8,6,4,3,5,7,9,2,10,1] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [9,5,4,6,3,7,2,8,10,1] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [9,6,4,3,5,7,2,8,10,1] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [9,6,5,7,3,2,4,8,10,1] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [9,7,4,3,5,2,6,8,10,1] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [9,7,5,3,2,4,6,8,10,1] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [10,5,4,6,3,7,2,8,1,9] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [10,6,4,3,5,7,2,8,1,9] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [10,6,5,7,3,2,4,8,1,9] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [10,7,4,3,5,2,6,8,1,9] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [10,7,5,3,2,4,6,8,1,9] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]
=> [10,8,4,3,5,2,6,1,7,9] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,3,2,4,6,1,7,9] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,4,6,2,1,3,7,9] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,3,2,4,1,5,7,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],.]]
=> [8,6,5,7,9,4,10,3,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],.]]
=> [9,6,5,7,4,8,10,3,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],.]]
=> [9,7,5,4,6,8,10,3,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [.,[[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],.]]
=> [10,6,5,7,4,8,3,9,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [.,[[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],.]]
=> [10,7,5,4,6,8,3,9,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [.,[[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],.]]
=> [10,8,5,4,6,3,7,9,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [.,[[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],.]]
=> [10,8,6,4,3,5,7,9,11,2,12,1] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [.,[[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],.]]
=> [11,6,5,7,4,8,3,9,2,10,12,1] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [.,[[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],.]]
=> [11,7,5,4,6,8,3,9,2,10,12,1] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [.,[[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],.]]
=> [11,8,5,4,6,3,7,9,2,10,12,1] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [.,[[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],.]]
=> [11,8,6,4,3,5,7,9,2,10,12,1] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [.,[[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],.]]
=> [11,9,5,4,6,3,7,2,8,10,12,1] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [.,[[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],.]]
=> [11,9,6,4,3,5,7,2,8,10,12,1] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [.,[[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],.]]
=> [11,9,7,4,3,5,2,6,8,10,12,1] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [.,[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [11,9,7,5,3,2,4,6,8,10,12,1] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],[.,.]]
=> [12,6,5,7,4,8,3,9,2,10,1,11] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],[.,.]]
=> [12,7,5,4,6,8,3,9,2,10,1,11] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],[.,.]]
=> [12,8,5,4,6,3,7,9,2,10,1,11] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],[.,.]]
=> [12,8,6,4,3,5,7,9,2,10,1,11] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],[.,.]]
=> [12,9,5,4,6,3,7,2,8,10,1,11] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],[.,.]]
=> [12,9,6,4,3,5,7,2,8,10,1,11] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],[.,.]]
=> [12,9,7,4,3,5,2,6,8,10,1,11] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [12,9,7,5,3,2,4,6,8,10,1,11] => ?
=> ? = 0 + 1
[[[[],[[],[[],[[],[]]]]],[]],[]]
=> [[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]]
=> [12,10,5,4,6,3,7,2,8,1,9,11] => ?
=> ? = 0 + 1
[[[[],[[],[[[],[]],[]]]],[]],[]]
=> [[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]]
=> [12,10,6,4,3,5,7,2,8,1,9,11] => ?
=> ? = 0 + 1
[[[[],[[[],[[],[]]],[]]],[]],[]]
=> [[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]]
=> [12,10,7,4,3,5,2,6,8,1,9,11] => ?
=> ? = 0 + 1
[[[[],[[[[],[]],[]],[]]],[]],[]]
=> [[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [12,10,7,5,3,2,4,6,8,1,9,11] => ?
=> ? = 0 + 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000319
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[]
=> []
=> [] => []
=> ? = 1 + 1
[[]]
=> [1,0]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 0 + 1
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,5,7,8,1,2,4,9,6,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,6,7,1,8,2,4,9,5,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,1,7,8,2,3,9,6,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,6,7,1,2,8,3,9,5,10] => ?
=> ? = 0 + 1
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,6,1,7,2,8,3,9,4,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,2,4,6,8,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,3,5,7,10,11,2,12,4,6,8,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,3,5,8,10,11,2,4,12,6,7,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,3,5,9,10,2,11,4,12,6,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,3,6,8,10,11,2,4,5,12,7,9] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,3,6,9,10,2,11,4,5,12,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,3,7,9,10,2,4,11,5,12,6,8] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,8,9,2,10,4,11,5,12,6,7] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,4,6,8,10,11,2,3,5,7,12,9] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,4,6,9,10,2,11,3,5,7,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,4,7,9,10,2,3,11,5,6,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,8,9,2,10,3,11,5,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> [1,5,7,9,10,2,3,4,11,6,12,8] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,8,9,2,10,3,4,11,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,8,9,2,3,10,4,11,5,12,7] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,7,8,2,9,3,10,4,11,5,12,6] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [2,4,6,8,10,11,1,3,5,7,9,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [2,4,6,9,10,1,11,3,5,7,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [2,4,7,9,10,1,3,11,5,6,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,8,9,1,10,3,11,5,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> [2,5,7,9,10,1,3,4,11,6,8,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,8,9,1,10,3,4,11,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,8,9,1,3,10,4,11,5,7,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,7,8,1,9,3,10,4,11,5,6,12] => ?
=> ? = 0 + 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[]
=> []
=> [] => []
=> ? = 1 + 1
[[]]
=> [1,0]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 0 + 1
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,5,7,8,1,2,4,9,6,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,6,7,1,8,2,4,9,5,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,1,7,8,2,3,9,6,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,6,7,1,2,8,3,9,5,10] => ?
=> ? = 0 + 1
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,6,1,7,2,8,3,9,4,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,2,4,6,8,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,3,5,7,10,11,2,12,4,6,8,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,3,5,8,10,11,2,4,12,6,7,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,3,5,9,10,2,11,4,12,6,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,3,6,8,10,11,2,4,5,12,7,9] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,3,6,9,10,2,11,4,5,12,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,3,7,9,10,2,4,11,5,12,6,8] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,8,9,2,10,4,11,5,12,6,7] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,4,6,8,10,11,2,3,5,7,12,9] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,4,6,9,10,2,11,3,5,7,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,4,7,9,10,2,3,11,5,6,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,8,9,2,10,3,11,5,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> [1,5,7,9,10,2,3,4,11,6,12,8] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,8,9,2,10,3,4,11,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,8,9,2,3,10,4,11,5,12,7] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,7,8,2,9,3,10,4,11,5,12,6] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [2,4,6,8,10,11,1,3,5,7,9,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [2,4,6,9,10,1,11,3,5,7,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [2,4,7,9,10,1,3,11,5,6,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,8,9,1,10,3,11,5,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> [2,5,7,9,10,1,3,4,11,6,8,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,8,9,1,10,3,4,11,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,8,9,1,3,10,4,11,5,7,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,7,8,1,9,3,10,4,11,5,6,12] => ?
=> ? = 0 + 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000392
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[]
=> []
=> [] => ? => ? = 1 + 1
[[]]
=> [1,0]
=> [1] => => ? = -1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => 0 => 0 = -1 + 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => 1 => 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = -1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = -1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = -1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1 = 0 + 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1 = 0 + 1
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ? => ? = 0 + 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ? => ? = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ? => ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ? => ? = 0 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ? => ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ? => ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ? => ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ? => ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ? => ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ? => ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ? => ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ? => ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ? => ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ? => ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ? => ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ? => ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ? => ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ? => ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ? => ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,5,7,8,1,2,4,9,6,10] => ? => ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,6,7,1,8,2,4,9,5,10] => ? => ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,1,7,8,2,3,9,6,10] => ? => ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,6,7,1,2,8,3,9,5,10] => ? => ? = 0 + 1
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,6,1,7,2,8,3,9,4,10] => ? => ? = 0 + 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,2,4,6,8,10] => ? => ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,3,5,7,10,11,2,12,4,6,8,9] => ? => ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,3,5,8,10,11,2,4,12,6,7,9] => ? => ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,3,5,9,10,2,11,4,12,6,7,8] => ? => ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,3,6,8,10,11,2,4,5,12,7,9] => ? => ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,3,6,9,10,2,11,4,5,12,7,8] => ? => ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,3,7,9,10,2,4,11,5,12,6,8] => ? => ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,8,9,2,10,4,11,5,12,6,7] => ? => ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,4,6,8,10,11,2,3,5,7,12,9] => ? => ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,4,6,9,10,2,11,3,5,7,12,8] => ? => ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,4,7,9,10,2,3,11,5,6,12,8] => ? => ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,8,9,2,10,3,11,5,6,12,7] => ? => ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> [1,5,7,9,10,2,3,4,11,6,12,8] => ? => ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,8,9,2,10,3,4,11,6,12,7] => ? => ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,8,9,2,3,10,4,11,5,12,7] => ? => ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,7,8,2,9,3,10,4,11,5,12,6] => ? => ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [2,4,6,8,10,11,1,3,5,7,9,12] => ? => ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [2,4,6,9,10,1,11,3,5,7,8,12] => ? => ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [2,4,7,9,10,1,3,11,5,6,8,12] => ? => ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,8,9,1,10,3,11,5,6,7,12] => ? => ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> [2,5,7,9,10,1,3,4,11,6,8,12] => ? => ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,8,9,1,10,3,4,11,6,7,12] => ? => ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,8,9,1,3,10,4,11,5,7,12] => ? => ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,7,8,1,9,3,10,4,11,5,6,12] => ? => ? = 0 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001587
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[]
=> []
=> [] => []
=> ? = 1 + 1
[[]]
=> [1,0]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 0 + 1
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,5,7,8,1,2,4,9,6,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,6,7,1,8,2,4,9,5,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,1,7,8,2,3,9,6,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,6,7,1,2,8,3,9,5,10] => ?
=> ? = 0 + 1
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,6,1,7,2,8,3,9,4,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,2,4,6,8,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,3,5,7,10,11,2,12,4,6,8,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,3,5,8,10,11,2,4,12,6,7,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,3,5,9,10,2,11,4,12,6,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,3,6,8,10,11,2,4,5,12,7,9] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,3,6,9,10,2,11,4,5,12,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,3,7,9,10,2,4,11,5,12,6,8] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,8,9,2,10,4,11,5,12,6,7] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,4,6,8,10,11,2,3,5,7,12,9] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,4,6,9,10,2,11,3,5,7,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,4,7,9,10,2,3,11,5,6,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,8,9,2,10,3,11,5,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> [1,5,7,9,10,2,3,4,11,6,12,8] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,8,9,2,10,3,4,11,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,8,9,2,3,10,4,11,5,12,7] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,7,8,2,9,3,10,4,11,5,12,6] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [2,4,6,8,10,11,1,3,5,7,9,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [2,4,6,9,10,1,11,3,5,7,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [2,4,7,9,10,1,3,11,5,6,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,8,9,1,10,3,11,5,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> [2,5,7,9,10,1,3,4,11,6,8,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,8,9,1,10,3,4,11,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,8,9,1,3,10,4,11,5,7,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,7,8,1,9,3,10,4,11,5,6,12] => ?
=> ? = 0 + 1
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001918
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[]
=> []
=> [] => []
=> ? = 1 + 1
[[]]
=> [1,0]
=> [1] => [1]
=> 0 = -1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0 = -1 + 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0 = -1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = -1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = -1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 0 + 1
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 0 + 1
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 0 + 1
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ? = 0 + 1
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ?
=> ? = 0 + 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ?
=> ? = 0 + 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ?
=> ? = 0 + 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ?
=> ? = 0 + 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ?
=> ? = 0 + 1
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ?
=> ? = 0 + 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ?
=> ? = 0 + 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ?
=> ? = 0 + 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ?
=> ? = 0 + 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ?
=> ? = 0 + 1
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,5,7,8,1,2,4,9,6,10] => ?
=> ? = 0 + 1
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,6,7,1,8,2,4,9,5,10] => ?
=> ? = 0 + 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,1,7,8,2,3,9,6,10] => ?
=> ? = 0 + 1
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,6,7,1,2,8,3,9,5,10] => ?
=> ? = 0 + 1
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,6,1,7,2,8,3,9,4,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,2,4,6,8,10] => ?
=> ? = 0 + 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,3,5,7,10,11,2,12,4,6,8,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,3,5,8,10,11,2,4,12,6,7,9] => ?
=> ? = 0 + 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,3,5,9,10,2,11,4,12,6,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,3,6,8,10,11,2,4,5,12,7,9] => ?
=> ? = 0 + 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,3,6,9,10,2,11,4,5,12,7,8] => ?
=> ? = 0 + 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,3,7,9,10,2,4,11,5,12,6,8] => ?
=> ? = 0 + 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,8,9,2,10,4,11,5,12,6,7] => ?
=> ? = 0 + 1
[[],[[[],[[],[[],[[],[]]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,4,6,8,10,11,2,3,5,7,12,9] => ?
=> ? = 0 + 1
[[],[[[],[[],[[[],[]],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,4,6,9,10,2,11,3,5,7,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[],[[],[]]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,4,7,9,10,2,3,11,5,6,12,8] => ?
=> ? = 0 + 1
[[],[[[],[[[[],[]],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,8,9,2,10,3,11,5,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[],[[],[[],[]]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0,0]
=> [1,5,7,9,10,2,3,4,11,6,12,8] => ?
=> ? = 0 + 1
[[],[[[[],[[[],[]],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,8,9,2,10,3,4,11,6,12,7] => ?
=> ? = 0 + 1
[[],[[[[[],[[],[]]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,8,9,2,3,10,4,11,5,12,7] => ?
=> ? = 0 + 1
[[],[[[[[[],[]],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,7,8,2,9,3,10,4,11,5,12,6] => ?
=> ? = 0 + 1
[[[],[[],[[],[[],[[],[]]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [2,4,6,8,10,11,1,3,5,7,9,12] => ?
=> ? = 0 + 1
[[[],[[],[[],[[[],[]],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [2,4,6,9,10,1,11,3,5,7,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[],[[],[]]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [2,4,7,9,10,1,3,11,5,6,8,12] => ?
=> ? = 0 + 1
[[[],[[],[[[[],[]],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,8,9,1,10,3,11,5,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[],[[],[[],[]]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> [2,5,7,9,10,1,3,4,11,6,8,12] => ?
=> ? = 0 + 1
[[[],[[[],[[[],[]],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,8,9,1,10,3,4,11,6,7,12] => ?
=> ? = 0 + 1
[[[],[[[[],[[],[]]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,8,9,1,3,10,4,11,5,7,12] => ?
=> ? = 0 + 1
[[[],[[[[[],[]],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,7,8,1,9,3,10,4,11,5,6,12] => ?
=> ? = 0 + 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000668The least common multiple of the parts of the partition. St000253The crossing number of a set partition. St000058The order of a permutation. St000535The rank-width of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001029The size of the core of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000254The nesting number of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St000730The maximal arc length of a set partition. St000919The number of maximal left branches of a binary tree. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000845The maximal number of elements covered by an element in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000640The rank of the largest boolean interval in a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000651The maximal size of a rise in a permutation. St000028The number of stack-sorts needed to sort a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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