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Your data matches 227 different statistics following compositions of up to 3 maps.
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Matching statistic: St001851
St001851: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[-1] => 1
[1,2] => 1
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 3
[2,1] => 1
[2,-1] => 0
[-2,1] => 0
[-2,-1] => 1
[1,2,3] => 1
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 3
[-1,2,3] => 1
[-1,2,-3] => 3
[-1,-2,3] => 5
[-1,-2,-3] => 13
[1,3,2] => 1
[1,3,-2] => 0
[1,-3,2] => 0
[1,-3,-2] => 1
[-1,3,2] => 1
[-1,3,-2] => 0
[-1,-3,2] => 0
[-1,-3,-2] => 1
[2,1,3] => 1
[2,1,-3] => 1
[2,-1,3] => 0
[2,-1,-3] => 0
[-2,1,3] => 0
[-2,1,-3] => 0
[-2,-1,3] => 3
[-2,-1,-3] => 3
[2,3,1] => 0
[2,3,-1] => 0
[2,-3,1] => 0
[2,-3,-1] => 0
[-2,3,1] => 0
[-2,3,-1] => 0
[-2,-3,1] => 0
[-2,-3,-1] => 0
[3,1,2] => 0
[3,1,-2] => 0
[3,-1,2] => 0
[3,-1,-2] => 0
[-3,1,2] => 0
[-3,1,-2] => 0
[-3,-1,2] => 0
[-3,-1,-2] => 0
Description
The number of Hecke atoms of a signed permutation.
For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set
$$
\{ w\in\mathfrak H_n | w^{-1} \star w = z\},
$$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 62%●distinct values known / distinct values provided: 10%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 62%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => [1] => ([],1)
=> ? ∊ {1,1}
[-1] => [1] => [1] => ([],1)
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,-2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,-2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,-1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,-1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[1,3,-2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[1,-3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[1,-3,-2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[-1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[-1,3,-2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[-1,-3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[-1,-3,-2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[2,1,-3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[2,-1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[2,-1,-3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[-2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[-2,1,-3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[-2,-1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[-2,-1,-3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,-1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,-1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,-2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,-2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,-1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,-1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[-1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,-2] => [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000929
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
St000929: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
Values
[1] => []
=> ? ∊ {1,1}
[-1] => [1]
=> ? ∊ {1,1}
[1,2] => []
=> ? ∊ {1,1,1,1,3}
[1,-2] => [1]
=> ? ∊ {1,1,1,1,3}
[-1,2] => [1]
=> ? ∊ {1,1,1,1,3}
[-1,-2] => [1,1]
=> 1
[2,1] => []
=> ? ∊ {1,1,1,1,3}
[2,-1] => [2]
=> 0
[-2,1] => [2]
=> 0
[-2,-1] => []
=> ? ∊ {1,1,1,1,3}
[1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,1]
=> 1
[-1,2,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,1]
=> 1
[-1,-2,3] => [1,1]
=> 1
[-1,-2,-3] => [1,1,1]
=> 1
[1,3,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [2]
=> 0
[1,-3,2] => [2]
=> 0
[1,-3,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [2,1]
=> 0
[-1,-3,2] => [2,1]
=> 0
[-1,-3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2]
=> 0
[2,-1,-3] => [2,1]
=> 0
[-2,1,3] => [2]
=> 0
[-2,1,-3] => [2,1]
=> 0
[-2,-1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [3]
=> 0
[2,-3,1] => [3]
=> 0
[2,-3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [3]
=> 0
[-2,3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [3]
=> 0
[3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3]
=> 0
[3,-1,2] => [3]
=> 0
[3,-1,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3]
=> 0
[-3,1,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3]
=> 0
[3,2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [2]
=> 0
[3,-2,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [2,1]
=> 0
[-3,2,1] => [2]
=> 0
[-3,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [2,1]
=> 0
[-3,-2,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,1]
=> 1
[1,-2,3,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,1]
=> 1
[1,-2,-3,4] => [1,1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> 1
[-1,2,3,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,3,-4] => [1,1]
=> 1
[-1,2,-3,4] => [1,1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> 1
[-1,-2,3,4] => [1,1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> 1
[1,2,4,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,-3] => [2]
=> 0
[1,2,-4,3] => [2]
=> 0
[1,2,-4,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,-3] => [2,1]
=> 0
[1,-2,-4,3] => [2,1]
=> 0
[1,-2,-4,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,-3] => [2,1]
=> 0
[-1,2,-4,3] => [2,1]
=> 0
[-1,2,-4,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-2,4,3] => [1,1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> 0
[-1,-2,-4,-3] => [1,1]
=> 1
[1,3,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-2,4] => [2]
=> 0
[1,3,-2,-4] => [2,1]
=> 0
[1,-3,-2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-3,-2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,3,2,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-3,-2,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,4,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-4,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000790
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 56%●distinct values known / distinct values provided: 30%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 56%●distinct values known / distinct values provided: 30%
Values
[1] => []
=> []
=> ? ∊ {1,1}
[-1] => [1]
=> [1,0]
=> ? ∊ {1,1}
[1,2] => []
=> []
=> ? ∊ {1,1,1,1,3}
[1,-2] => [1]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,2] => [1]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => []
=> []
=> ? ∊ {1,1,1,1,3}
[2,-1] => [2]
=> [1,0,1,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> 0
[-2,-1] => []
=> []
=> ? ∊ {1,1,1,1,3}
[1,2,3] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [2]
=> [1,0,1,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> 0
[1,-3,-2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-1,-3,-2] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2]
=> [1,0,1,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,1,3] => [2]
=> [1,0,1,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,-1,3] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,-3,-1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[-2,3,-1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,-1,-2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[-3,1,-2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,2,1] => [2]
=> [1,0,1,0]
=> 0
[-3,2,-1] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,-2,-1] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-1,2,3,4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> 0
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> 0
[1,2,-4,-3] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,-2,-4,-3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-1,2,-4,-3] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,2,-4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> 0
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,-3,-2,4] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-3,-2,-4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,3,2,4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-3,-2,4] => [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,4,2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-4,-2] => []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
Description
The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000683
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 56%●distinct values known / distinct values provided: 30%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 56%●distinct values known / distinct values provided: 30%
Values
[1] => []
=> []
=> []
=> ? ∊ {1,1}
[-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1}
[1,2] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[2,1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,-1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,2,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,-3,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,-1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,-3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[-2,3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,-1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[-3,1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-3,2,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,2,-4,-3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,3,2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-3,-2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-3,-2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,3,2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-3,-2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,4,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-4,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
Matching statistic: St001139
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
Values
[1] => []
=> []
=> []
=> ? ∊ {1,1}
[-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1}
[1,2] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[2,1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,-1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,2,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,-3,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,-1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,-3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[-2,3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,-1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[-3,1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[-3,2,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,2,-4,-3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,3,2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-3,-2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-3,-2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,3,2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-3,-2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,4,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-4,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St001204
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 56%●distinct values known / distinct values provided: 20%
Values
[1] => []
=> []
=> []
=> ? ∊ {1,1}
[-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1}
[1,2] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1,1,1,3}
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,-1] => []
=> []
=> []
=> ? ∊ {1,1,1,1,3}
[1,2,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,-3,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-2,-1,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,3,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,1,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-3,2,-1] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,-4,-3] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[-1,2,-4,-3] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,3,2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,-3,-2,4] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,-3,-2,-4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,3,2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[-1,-3,-2,4] => [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,4,2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
[1,3,-4,-2] => []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,9,9,13,13,15,19,19,85}
Description
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St000512
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 52%●distinct values known / distinct values provided: 20%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 52%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000936
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => [1] => [1]
=> []
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[-1,-2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[-2,-1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-1,-3,-2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-1,-3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-2,-3,-1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-1,-2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[-3,-2,-1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,5,13}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
The following 217 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000219The number of occurrences of the pattern 231 in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000053The number of valleys of the Dyck path. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000628The balance of a binary word. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000869The sum of the hook lengths of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000976The sum of the positions of double up-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000992The alternating sum of the parts of an integer partition. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001214The aft of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001910The height of the middle non-run of a Dyck path. St001930The weak major index of a binary word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001956The comajor index for set-valued two-row standard Young tableaux. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras.
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