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Your data matches 613 different statistics following compositions of up to 3 maps.
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Matching statistic: St000824
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(load all 254 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000824: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000824: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => 4 = 3 + 1
Description
The sum of the number of descents and the number of recoils of a permutation.
This statistic is the sum of [[St000021]] and [[St000354]].
Matching statistic: St000896
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000896: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [[0,1],[1,0]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
Description
The number of zeros on the main diagonal of an alternating sign matrix.
Matching statistic: St001299
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001299: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001299: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
Description
The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.
Matching statistic: St001166
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001166: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001166: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 5 = 3 + 2
Description
Number 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.
Matching statistic: St001182
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 3 + 2
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000016
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [[1,4],[2],[3],[5],[6]]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [[1,4],[2],[3],[5],[6]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [[1,5],[2],[3],[4],[6]]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [[1,2],[3,5],[4],[6]]
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [[1,5],[2],[3],[4],[6]]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [[1,5],[2],[3],[4],[6]]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [[1,4],[2,6],[3],[5]]
=> 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [[1,4],[2,6],[3],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [[1,6],[2],[3],[4],[5]]
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [[1,2],[3,6],[4],[5]]
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [[1,3],[2,6],[4],[5]]
=> 3
Description
The number of attacking pairs of a standard tableau.
Note that this is actually a statistic on the underlying partition.
A pair of cells $(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds:
1. $c$ and $d$ lie in the same row with $c$ strictly to the west of $d$.
2. $c$ is in the row immediately to the south of $d$, and $c$ lies strictly east of $d$.
Matching statistic: St000154
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => 3
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
Matching statistic: St000185
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,2] => [1,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [5,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [5,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,1]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [4,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [5,1]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [5,1]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [5,1]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [4,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [4,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [4,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [4,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [4,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [5,1]
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [4,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [4,1,1]
=> 3
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St000421
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
Description
The number of Dyck paths that are weakly below a Dyck path, except for the path itself.
Matching statistic: St000472
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,1,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,1,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,1,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,5,4,2,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1,5,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,2,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,1,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,1,2] => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,1,3,2] => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,2,1,3] => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,1,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,5,1,4,2,3] => 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,5,3,1,4,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [6,5,3,2,1,4] => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [6,1,5,4,3,2] => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [6,1,5,4,2,3] => 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [6,1,5,2,4,3] => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,1,5,3,2,4] => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [6,4,1,5,3,2] => 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [6,4,1,5,2,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [6,4,3,1,5,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [6,4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,6,5,4,2,3] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,6,5,2,4,3] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,6,5,3,2,4] => 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,6,2,5,4,3] => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,6,4,2,5,3] => 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [5,1,6,4,3,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [5,1,6,4,2,3] => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [5,1,6,2,4,3] => 3
Description
The sum of the ascent bottoms of a permutation.
The following 603 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001726The number of visible inversions of a permutation. St001961The sum of the greatest common divisors of all pairs of parts. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000071The number of maximal chains in a poset. St000082The number of elements smaller than a binary tree in Tamari order. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000366The number of double descents of a permutation. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000673The number of non-fixed points of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000828The spearman's rho of a permutation and the identity permutation. St000830The total displacement of a permutation. St000883The number of longest increasing subsequences of a permutation. St000912The number of maximal antichains in a poset. St000933The number of multipartitions of sizes given by an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001346The number of parking functions that give the same permutation. St001458The rank of the adjacency matrix of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001902The number of potential covers of a poset. St001959The product of the heights of the peaks of a Dyck path. St000037The sign of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000004The major index of a permutation. St000012The area of a Dyck path. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000141The maximum drop size of a permutation. St000156The Denert index of a permutation. St000169The cocharge of a standard tableau. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000330The (standard) major index of a standard tableau. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000336The leg major index of a standard tableau. St000354The number of recoils of a permutation. St000391The sum of the positions of the ones in a binary word. St000460The hook length of the last cell along the main diagonal of an integer partition. St000490The intertwining number of a set partition. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000662The staircase size of the code of a permutation. 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. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000742The number of big ascents of a permutation after prepending zero. St000762The sum of the positions of the weak records of an integer composition. St000792The Grundy value for the game of ruler on a binary word. St000796The stat' of a permutation. St000798The makl of a permutation. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000829The Ulam distance of a permutation to the identity permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001175The size of a partition minus the hook length of the base cell. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001489The maximum of the number of descents and the number of inverse descents. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001564The value of the forgotten symmetric functions when all variables set to 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001651The Frankl number of a lattice. St001697The shifted natural comajor index of a standard Young tableau. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001846The number of elements which do not have a complement in the lattice. St001874Lusztig's a-function for the symmetric group. St000024The number of double up and double down steps of a Dyck path. St000039The number of crossings of a permutation. St000053The number of valleys of the Dyck path. St000100The number of linear extensions of a poset. St000148The number of odd parts of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000232The number of crossings of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000325The width of the tree associated to a permutation. St000343The number of spanning subgraphs of a graph. St000350The sum of the vertex degrees of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000422The energy of a graph, if it is integral. St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000465The first Zagreb index of a graph. St000468The Hosoya index of a graph. St000470The number of runs in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000571The F-index (or forgotten topological index) of a graph. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000638The number of up-down runs of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000972The composition number of a graph. St000992The alternating sum of the parts of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000995The largest even part of an integer partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001459The number of zero columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001486The number of corners of the ribbon associated with an integer composition. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001623The number of doubly irreducible elements of a lattice. St001657The number of twos in an integer partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001679The number of subsets of a lattice whose meet is the bottom element. St001727The number of invisible inversions of a permutation. St001735The number of permutations with the same set of runs. St001754The number of tolerances of a finite lattice. St001840The number of descents of a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000058The order of a permutation. St000146The Andrews-Garvan crank of a partition. St000187The determinant of an alternating sign matrix. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001619The number of non-isomorphic sublattices of a lattice. St001625The Möbius invariant of a lattice. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000885The number of critical steps in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000646The number of big ascents of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000670The reversal length of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St001472The permanent of the Coxeter matrix of the poset. St000142The number of even parts of a partition. St000831The number of indices that are either descents or recoils. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001330The hat guessing number of a graph. St001964The interval resolution global dimension of a poset. St000809The reduced reflection length of the permutation. St001537The number of cyclic crossings of a permutation. St000485The length of the longest cycle of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000878The number of ones minus the number of zeros of a binary word. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001684The reduced word complexity of a permutation. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. St000847The number of standard Young tableaux whose descent set is the binary word. St001272The number of graphs with the same degree sequence. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000216The absolute length of a permutation. St000223The number of nestings in the permutation. St000331The number of upper interactions of a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000889The number of alternating sign matrices with the same antidiagonal sums. St000983The length of the longest alternating subword. St001077The prefix exchange distance of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St000015The number of peaks of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000722The number of different neighbourhoods in a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000259The diameter of a connected graph. St000741The Colin de Verdière graph invariant. St001545The second Elser number of a connected graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000264The girth of a graph, which is not a tree. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001875The number of simple modules with projective dimension at most 1. St000477The weight of a partition according to Alladi. 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. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000753The Grundy value for the game of Kayles on a binary word. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000997The even-odd crank of an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000928The sum of the coefficients 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000068The number of minimal elements in a poset. St000750The number of occurrences of the pattern 4213 in a permutation. St000288The number of ones in a binary word. St000454The largest eigenvalue of a graph if it is integral. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St000806The semiperimeter of the associated bargraph. St001817The number of flag weak exceedances of a signed permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St000313The number of degree 2 vertices of a graph. St001571The Cartan determinant of the integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000902 The minimal number of repetitions of an integer composition. St001524The degree of symmetry of a binary word. St000899The maximal number of repetitions of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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 $K[x]/(x^n)$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001675The number of parts equal to the part in the reversed composition. St001768The number of reduced words of a signed permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000295The length of the border of a binary word. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000489The number of cycles of a permutation of length at most 3. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000669The number of permutations obtained by switching ascents or descents of size 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001705The number of occurrences of the pattern 2413 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001866The nesting alignments of a signed permutation. St001948The number of augmented double ascents of a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000357The number of occurrences of the pattern 12-3. St000502The number of successions of a set partitions. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St001267The length of the Lyndon factorization of the binary word. St000248The number of anti-singletons of a set partition. St000904The maximal number of repetitions of an integer composition. St001060The distinguishing index of a graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000617The number of global maxima of a Dyck path. 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. St000907The number of maximal antichains of minimal length in a poset. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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 $eA$. 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. St001301The first Betti number of the order complex associated with the poset. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001720The minimal length of a chain of small intervals in a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001845The number of join irreducibles minus the rank of a lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. 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 $eA$. St000069The number of maximal elements of a poset. St000632The jump number of the poset. St000911The number of maximal antichains of maximal size in a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001429The number of negative entries in a signed permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001613The binary logarithm of the size of the center of a lattice. St001864The number of excedances of a signed permutation. St001881The number of factors of a lattice as a Cartesian product of lattices. St000022The number of fixed points of a permutation. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000417The size of the automorphism group of the ordered tree. St000451The length of the longest pattern of the form k 1 2. St000546The number of global descents of a permutation. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000679The pruning number of an ordered tree. St000731The number of double exceedences of a permutation. St000842The breadth of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001058The breadth of the ordered tree. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000307The number of rowmotion orbits of a poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000383The last part of an integer composition. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000871The number of very big ascents of a permutation. St000925The number of topologically connected components of a set partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001050The number of terminal closers of a set partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000233The number of nestings of a set partition. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000396The register function (or Horton-Strahler number) of a binary tree. 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. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. 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. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000701The protection number of a binary tree. St000748The major index of the permutation obtained by flattening the set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000862The number of parts of the shifted shape of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000981The length of the longest zigzag subpath. 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. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St000093The cardinality of a maximal independent set of vertices of a graph. St000385The number of vertices with out-degree 1 in a binary tree. St000444The length of the maximal rise of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000007The number of saliances of the permutation. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000119The number of occurrences of the pattern 321 in a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001820The size of the image of the pop stack sorting operator. St000891The number of distinct diagonal sums of a permutation matrix. St000028The number of stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St000153The number of adjacent cycles of a permutation. St000352The Elizalde-Pak rank of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000627The exponent of a binary word. St000729The minimal arc length of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000445The number of rises of length 1 of 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. St000464The Schultz index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000629The defect of a binary word. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000920The logarithmic height of a Dyck path. St001075The minimal size of a block of a set partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001645The pebbling number of a connected graph. St001851The number of Hecke atoms of a signed permutation. St000098The chromatic number of a graph. St000302The determinant of the distance matrix of a connected graph. St000982The length of the longest constant subword. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000402Half the size of the symmetry class of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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