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Your data matches 611 different statistics following compositions of up to 3 maps.
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Matching statistic: St000022
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Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of fixed points of a permutation.
Matching statistic: St000031
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(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000153
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(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000308
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000461
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Matching statistic: St000488
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000488: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000488: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of cycles of a permutation of length at most 2.
Matching statistic: St000489
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of cycles of a permutation of length at most 3.
Matching statistic: St000625
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000625: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000625: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i)
$$
This statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently coincides with [2], which satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0.
$$
Matching statistic: St000654
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000702
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 3
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => 3
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
The following 601 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001074The number of inversions of the cyclic embedding of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000441The number of successions of a permutation. St000520The number of patterns in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000989The number of final rises of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001405The number of bonds in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000203The number of external nodes of a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000228The size of a partition. St000247The number of singleton blocks of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000294The number of distinct factors of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000518The number of distinct subsequences in a binary word. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000636The hull number of a graph. St000657The smallest part of an integer composition. St000674The number of hills of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000734The last entry in the first row of a standard tableau. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000825The sum of the major and the inverse major index of a permutation. St000839The largest opener of a set partition. St000867The sum of the hook lengths in the first row of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000993The multiplicity of the largest part of an integer partition. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001127The sum of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001441The number of non-empty connected induced subgraphs of a graph. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001492The 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. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001933The largest multiplicity of a part in an integer partition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000064The number of one-box pattern of a permutation. St000070The number of antichains in a poset. St000108The number of partitions contained in the given partition. St000141The maximum drop size of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000214The number of adjacencies of a permutation. St000221The number of strong fixed points of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000246The number of non-inversions of a permutation. St000288The number of ones in a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000338The number of pixed points of a permutation. St000377The dinv defect of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000385The number of vertices with out-degree 1 in a binary tree. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000463The number of admissible inversions of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000532The total number of rook placements on a Ferrers board. St000546The number of global descents of a permutation. St000627The exponent of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000778The metric dimension of a graph. St000806The semiperimeter of the associated bargraph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001267The length of the Lyndon factorization of the binary word. St001340The cardinality of a minimal non-edge isolating set of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001759The Rajchgot index of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000060The greater neighbor of the maximum. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000295The length of the border of a binary word. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000519The largest length of a factor maximising the subword complexity. St000731The number of double exceedences of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001083The number of boxed occurrences of 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000293The number of inversions of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000921The number of internal inversions of a binary word. St001658The total number of rook placements on a Ferrers board. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001516The number of cyclic bonds of a permutation. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000050The depth or height of a binary tree. St000087The number of induced subgraphs. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000226The convexity of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000241The number of cyclical small excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000287The number of connected components of a graph. St000305The inverse major index of a permutation. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000446The disorder of a permutation. St000470The number of runs in a permutation. St000471The sum of the ascent tops of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000656The number of cuts of a poset. St000673The number of non-fixed points of a permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000990The first ascent of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001430The number of positive entries in a signed permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001519The pinnacle sum of a permutation. St001530The depth of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001622The number of join-irreducible elements of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St001828The Euler characteristic of a graph. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000104The number of facets in the order polytope of this poset. St000133The "bounce" of a permutation. St000151The number of facets in the chain polytope of the poset. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000451The length of the longest pattern of the form k 1 2. St000462The major index minus the number of excedences of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. 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). St001077The prefix exchange distance of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001220The width of a permutation. 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. 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. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001246The maximal difference between two consecutive entries of a permutation. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001468The smallest fixpoint of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001664The number of non-isomorphic subposets of a poset. St001726The number of visible inversions of a permutation. St001782The order of rowmotion on the set of order ideals of a poset. St001925The minimal number of zeros in a row of an alternating sign matrix. St001958The degree of the polynomial interpolating the values of a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000538The number of even inversions of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001377The major index minus the number of inversions of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001727The number of invisible inversions of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001948The number of augmented double ascents of a permutation. St000358The number of occurrences of the pattern 31-2. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000947The major index east count of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001808The box weight or horizontal decoration of a Dyck path. St000369The dinv deficit of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000134The size of the orbit of an alternating sign matrix under gyration. St000197The number of entries equal to positive one in the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. 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. 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. 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. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000327The number of cover relations in a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001480The number of simple summands of the module J^2/J^3. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001684The reduced word complexity of a permutation. St001834The number of non-isomorphic minors of a graph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000135The number of lucky cars of the parking function. St000942The number of critical left to right maxima of the parking functions. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001903The number of fixed points of a parking function. St001904The length of the initial strictly increasing segment of a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001937The size of the center of a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001621The number of atoms of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001615The number of join prime elements of a lattice. St001616The number of neutral elements in a lattice. St001617The dimension of the space of valuations of a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001846The number of elements which do not have a complement in the lattice. St001875The number of simple modules with projective dimension at most 1. St001881The number of factors of a lattice as a Cartesian product of lattices. St001625The Möbius invariant of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001857The number of edges in the reduced word graph of a signed permutation. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element.
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