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Your data matches 124 different statistics following compositions of up to 3 maps.
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Matching statistic: St000220
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,3,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[3,1,2,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000425
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,3,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[3,1,2,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000433
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000433: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000433: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,3,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[3,1,2,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.
Matching statistic: St000457
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [2,3,1] => 0
[3,1,2] => [3,2,1] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[2,4,3,1] => [4,3,2,1] => [2,3,4,1] => 0
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[3,4,1,2] => [4,3,2,1] => [2,3,4,1] => 0
[3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[4,2,1,3] => [4,3,2,1] => [2,3,4,1] => 0
[4,2,3,1] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,1,2] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,4,5,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[2,5,3,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[2,5,4,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,4,5,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,4,5,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,5,1,4,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,5,2,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,5,4,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[3,5,4,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,2,5,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,2,5,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,3,5,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,3,5,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,1,2,3] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,1,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,2,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,2,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,3,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[4,5,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[5,2,3,1,4] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[5,2,3,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[5,2,4,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[5,2,4,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[5,3,1,2,4] => [5,4,3,2,1] => [2,3,4,5,1] => 0
Description
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation.
According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.
Matching statistic: St001633
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,1,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000003
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1 = 0 + 1
[1,2] => [1,2] => [2]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,1,1]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [1,1,1]
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [3,1]
=> 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,2,1,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000078
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000078: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000078: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
Description
The number of alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
Matching statistic: St000255
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000255: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000255: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation π∈Sn+1 as a reduced subword of sn(sn−1sn)(sn−2sn−1sn)⋯(s1⋯sn), or the number of reduced pipe dreams for π.
Matching statistic: St000277
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [2] => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5] => 1 = 0 + 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
Description
The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a 2×2 square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition c=c1,…,ck of n, the number of ribbon shaped standard tableaux equals
\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},
where the sum is over all coarsenings of c obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]
The following 114 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000909The number of maximal chains of maximal size in a poset. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000057The Shynar inversion number of a standard tableau. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000218The number of occurrences of the pattern 213 in a permutation. St000225Difference between largest and smallest parts in a partition. St000359The number of occurrences of the pattern 23-1. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000682The Grundy value of Welter's game on a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001377The major index minus the number of inversions of a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001810The number of fixed points of a permutation smaller than its largest moved point. St000047The number of standard immaculate tableaux of a given shape. St000075The orbit size of a standard tableau under promotion. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St001313The number of Dyck paths above the lattice path given by a binary word. St001595The number of standard Young tableaux of the skew partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000293The number of inversions of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St000290The major index of a binary word. St000369The dinv deficit of a Dyck path. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001485The modular major index of a binary word. St001910The height of the middle non-run of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000048The multinomial of the parts of a partition. St000452The number of distinct eigenvalues of a graph. St000530The number of permutations with the same descent word as the given permutation. St000548The number of different non-empty partial sums of an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001128The exponens consonantiae of a partition. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000567The sum of the products of all pairs of parts. 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000222The number of alignments in the permutation. St001510The number of self-evacuating linear extensions of a finite poset. St000217The number of occurrences of the pattern 312 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000001The number of reduced words for a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000988The orbit size of a permutation under Foata's bijection. St000219The number of occurrences of the pattern 231 in a permutation. St000045The number of linear extensions of a binary tree. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000464The Schultz index of a connected graph. St001651The Frankl number of a lattice. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000881The number of short braid edges in the graph of braid moves of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern.
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