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Your data matches 112 different statistics following compositions of up to 3 maps.
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Matching statistic: St000884
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 0
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 0
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001125
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001216
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.
Matching statistic: St001230
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001274
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St001276
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St001657
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [2] => [2]
=> 1
[2,1] => [1,1] => [1,1]
=> 0
[1,2,3] => [3] => [3]
=> 0
[1,3,2] => [2,1] => [2,1]
=> 1
[2,1,3] => [1,2] => [2,1]
=> 1
[2,3,1] => [2,1] => [2,1]
=> 1
[3,1,2] => [1,2] => [2,1]
=> 1
[3,2,1] => [1,1,1] => [1,1,1]
=> 0
[1,2,3,4] => [4] => [4]
=> 0
[1,2,4,3] => [3,1] => [3,1]
=> 0
[1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,4,2] => [3,1] => [3,1]
=> 0
[1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,3,2] => [2,1,1] => [2,1,1]
=> 1
[2,1,3,4] => [1,3] => [3,1]
=> 0
[2,1,4,3] => [1,2,1] => [2,1,1]
=> 1
[2,3,1,4] => [2,2] => [2,2]
=> 2
[2,3,4,1] => [3,1] => [3,1]
=> 0
[2,4,1,3] => [2,2] => [2,2]
=> 2
[2,4,3,1] => [2,1,1] => [2,1,1]
=> 1
[3,1,2,4] => [1,3] => [3,1]
=> 0
[3,1,4,2] => [1,2,1] => [2,1,1]
=> 1
[3,2,1,4] => [1,1,2] => [2,1,1]
=> 1
[3,2,4,1] => [1,2,1] => [2,1,1]
=> 1
[3,4,1,2] => [2,2] => [2,2]
=> 2
[3,4,2,1] => [2,1,1] => [2,1,1]
=> 1
[4,1,2,3] => [1,3] => [3,1]
=> 0
[4,1,3,2] => [1,2,1] => [2,1,1]
=> 1
[4,2,1,3] => [1,1,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,2,1] => [2,1,1]
=> 1
[4,3,1,2] => [1,1,2] => [2,1,1]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [5] => [5]
=> 0
[1,2,3,5,4] => [4,1] => [4,1]
=> 0
[1,2,4,3,5] => [3,2] => [3,2]
=> 1
[1,2,4,5,3] => [4,1] => [4,1]
=> 0
[1,2,5,3,4] => [3,2] => [3,2]
=> 1
[1,2,5,4,3] => [3,1,1] => [3,1,1]
=> 0
[1,3,2,4,5] => [2,3] => [3,2]
=> 1
[1,3,2,5,4] => [2,2,1] => [2,2,1]
=> 2
[1,3,4,2,5] => [3,2] => [3,2]
=> 1
[1,3,4,5,2] => [4,1] => [4,1]
=> 0
[1,3,5,2,4] => [3,2] => [3,2]
=> 1
[1,3,5,4,2] => [3,1,1] => [3,1,1]
=> 0
[1,4,2,3,5] => [2,3] => [3,2]
=> 1
[1,4,2,5,3] => [2,2,1] => [2,2,1]
=> 2
[1,4,3,2,5] => [2,1,2] => [2,2,1]
=> 2
[1,4,3,5,2] => [2,2,1] => [2,2,1]
=> 2
[1,4,5,2,3] => [3,2] => [3,2]
=> 1
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000214
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001189
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000011The number of touch points (or returns) of a Dyck path. 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. St000658The number of rises of length 2 of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St001061The number of indices that are both descents and recoils of a permutation. St000993The multiplicity of the largest part of an integer partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St000946The sum of the skew hook positions in a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000260The radius of a connected graph. St000137The Grundy value of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000941The number of characters of the symmetric group whose value on the partition is even. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the 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. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000648The number of 2-excedences of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000259The diameter of a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000891The number of distinct diagonal sums of a permutation matrix. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice.
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