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Your data matches 168 different statistics following compositions of up to 3 maps.
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Matching statistic: St001211
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 2
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 5
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 2
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 5
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 4 = 5 - 1
Description
The length of the partition.
Matching statistic: St000507
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: 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
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [[1,2],[3]]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [[1,2],[3]]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [[1,3],[2]]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 4 = 5 - 1
Description
The number of ascents of a standard tableau.
Entry i of a standard Young tableau is an '''ascent''' if i+1 appears to the right or above i in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001250
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 4 = 5 - 1
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000393
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => 10 => 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 10011 => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 10011 => 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 10011 => 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => 101111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => 101110 => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => 101101 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => 101101 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => 101100 => 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => 101011 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => 101010 => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => 101011 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => 101011 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => 101010 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => 101001 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => 101001 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => 101001 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => 101000 => 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => 100111 => 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => 100110 => 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => 100101 => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => 100101 => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => 100100 => 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => 100111 => 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => 100110 => 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => 100111 => 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1] => 100111 => 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => 100110 => 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => 100101 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => 100101 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => 100101 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => 100100 => 5
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number α(G) of G.
Matching statistic: St000105
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 5 - 1
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] S2(n,k) given by the number of [[SetPartitions|set partitions]] of {1,…,n} into k blocks, see [1].
Matching statistic: St000147
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2,2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 10 => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2]
=> 100 => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 110 => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1010 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1010 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1010 => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1010 => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1110 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 1100 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 1100 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 11110 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 11010 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 101110 => 4 = 5 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000378The diagonal inversion number of an integer partition. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000925The number of topologically connected components of a set partition. St001286The annihilation number of a graph. St001462The number of factors of a standard tableaux under concatenation. St000482The (zero)-forcing number of a graph. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000245The number of ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000167The number of leaves of an ordered tree. St000308The height of the tree associated to a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000214The number of adjacencies of a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000007The number of saliances of the permutation. St000662The staircase size of the code of a permutation. St000306The bounce count of a Dyck path. St000553The number of blocks of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000062The length of the longest increasing subsequence of the permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. 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. St000015The number of peaks of a Dyck path. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001480The number of simple summands of the module J^2/J^3. St001321The number of vertices of the largest induced subforest of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001427The number of descents of a signed permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001405The number of bonds in a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000454The largest eigenvalue of a graph if it is integral. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000056The decomposition (or block) number of a permutation. St000144The pyramid weight of the Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000240The number of indices that are not small excedances. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000331The number of upper interactions of a Dyck path. St000619The number of cyclic descents of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001083The number of boxed occurrences of 132 in a permutation. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001432The order dimension of the partition. St001863The number of weak excedances of a signed permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001896The number of right descents of a signed permutations. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000871The number of very big ascents of a permutation. St001626The number of maximal proper sublattices of a lattice. St001935The number of ascents in a parking function. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000942The number of critical left to right maxima of the parking functions. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000292The number of ascents of a binary word. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.
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