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Your data matches 670 different statistics following compositions of up to 3 maps.
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Matching statistic: St000951
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000951: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000951: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0
Description
The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.
Matching statistic: St000687
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St000687: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 0
[]
=> ? = 1
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St001490
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 62%●distinct values known / distinct values provided: 33%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 62%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1],[]]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[]
=> []
=> []
=> ?
=> ? = 1 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St000181
(load all 110 compositions to match this statistic)
(load all 110 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[]
=> []
=> ?
=> ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001868
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,4,2,1] => [3,4,2,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => [1,4,5,3,2] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,3,5,2] => [1,4,3,5,2] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 0
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[]
=> [] => [] => [] => ? = 1
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St001882
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [3,2,1] => 0
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[]
=> [] => [] => [] => ? = 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001890
(load all 130 compositions to match this statistic)
(load all 130 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 54%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ? = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[]
=> []
=> ?
=> ? = 1 + 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001866
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[]
=> [] => [] => [] => ? = 1
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
* $-i < -j < -\pi(j) < -\pi(i)$, or
* $-i < j \leq \pi(j) < -\pi(i)$, or
* $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St001435
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 51%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 51%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 0
[]
=> []
=> []
=> [[],[]]
=> ? = 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 51%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 51%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 0
[]
=> []
=> []
=> [[],[]]
=> ? = 1
Description
The number of missing boxes of a skew partition.
The following 660 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001487The number of inner corners of a skew partition. St000264The girth of a graph, which is not a tree. St000068The number of minimal elements in a poset. St001330The hat guessing number of a graph. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000842The breadth of a permutation. St001875The number of simple modules with projective dimension at most 1. St000488The number of cycles of a permutation of length at most 2. St001301The first Betti number of the order complex associated with the poset. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001867The number of alignments of type EN of a signed permutation. St000335The difference of lower and upper interactions. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000629The defect of a binary word. St000877The depth of the binary word interpreted as a path. St001811The Castelnuovo-Mumford regularity of a permutation. St000456The monochromatic index of a connected graph. St000805The number of peaks of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000902 The minimal number of repetitions of an integer composition. St000914The sum of the values of the Möbius function of a poset. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001545The second Elser number of a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000731The number of double exceedences of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001396Number of triples of incomparable elements in a finite poset. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St000925The number of topologically connected components of a set partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000069The number of maximal elements of a poset. St000383The last part of an integer composition. St000441The number of successions of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000563The number of overlapping pairs of blocks of a set partition. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000665The number of rafts of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001394The genus of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000696The number of cycles in the breakpoint graph of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001669The number of single rises in a Dyck path. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000678The number of up steps after the last double rise of a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000981The length of the longest zigzag subpath. St001964The interval resolution global dimension of a poset. St000534The number of 2-rises of a permutation. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000627The exponent of a binary word. St000907The number of maximal antichains of minimal length in a poset. St001060The distinguishing index of a graph. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000648The number of 2-excedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001851The number of Hecke atoms of a signed permutation. St000141The maximum drop size of a permutation. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000002The number of occurrences of the pattern 123 in a permutation. St000359The number of occurrences of the pattern 23-1. St000386The number of factors DDU in a Dyck path. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001115The number of even descents of a permutation. St000115The single entry in the last row. St000124The cardinality of the preimage of the Simion-Schmidt map. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001903The number of fixed points of a parking function. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000065The number of entries equal to -1 in an alternating sign matrix. 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. St000233The number of nestings of a set partition. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. 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. St000895The number of ones on the main diagonal of an alternating sign matrix. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St000948The chromatic discriminant of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001383The BG-rank of an integer partition. St001434The number of negative sum pairs of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001947The number of ties in a parking function. St001961The sum of the greatest common divisors of all pairs of parts. St000028The number of stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St000284The Plancherel distribution on integer partitions. St000296The length of the symmetric border of a binary word. St000618The number of self-evacuating tableaux of given shape. St000632The jump number of the poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001427The number of descents of a signed permutation. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000527The width of the poset. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001928The number of non-overlapping descents in a permutation. St000058The order of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000077The number of boxed and circled entries. St000090The variation of a composition. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000133The "bounce" of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000232The number of crossings of a set partition. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000268The number of strongly connected orientations of a graph. St000274The number of perfect matchings of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000338The number of pixed points of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000461The rix statistic of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000552The number of cut vertices of a graph. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000637The length of the longest cycle in a graph. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000680The Grundy value for Hackendot on posets. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001008Number of indecomposable injective modules with projective dimension 1 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001281The normalized isoperimetric number of a graph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001377The major index minus the number of inversions of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001411The number of patterns 321 or 3412 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001596The number of two-by-two squares inside a skew partition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001691The number of kings in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001727The number of invisible inversions of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001736The total number of cycles in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001783The number of odd automorphisms of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001822The number of alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000037The sign of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000472The sum of the ascent bottoms of a permutation. St000492The rob statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000529The number of permutations whose descent word is the given binary word. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000640The rank of the largest boolean interval in a poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000823The number of unsplittable factors of the set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000886The number of permutations with the same antidiagonal sums. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001081The number of minimal length factorizations of a permutation into star transpositions. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001153The number of blocks with even minimum in a set partition. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001277The degeneracy of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001468The smallest fixpoint of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001597The Frobenius rank of a skew partition. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001768The number of reduced words of a signed permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001792The arboricity of a graph. St001807The lower middle entry of a permutation. St001850The number of Hecke atoms of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000469The distinguishing number of a graph. St000489The number of cycles of a permutation of length at most 3. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000839The largest opener of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000917The open packing number of a graph. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001316The domatic number of a graph. St001399The distinguishing number of a poset. St001433The flag major index of a signed permutation. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001716The 1-improper chromatic number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001817The number of flag weak exceedances of a signed permutation. St001819The flag Denert index of a signed permutation. St001826The maximal number of leaves on a vertex of a graph. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000230Sum of the minimal elements of the blocks of a set partition. St001108The 2-dynamic chromatic number of a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000219The number of occurrences of the pattern 231 in a permutation. St001556The number of inversions of the third entry of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000353The number of inner valleys of a permutation.
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