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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000362
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St001176
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 1
[1,2,3] => [3]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [2,1]
=> 1
[3,1,2] => [2,1]
=> 1
[3,2,1] => [1,1,1]
=> 2
[1,2,3,4] => [4]
=> 0
[1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [3,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 2
[2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 1
[1,2,4,3,5] => [4,1]
=> 1
[1,2,4,5,3] => [4,1]
=> 1
[1,2,5,3,4] => [4,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> 1
[1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [4,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [3,2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> 1
[1,4,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [3,2]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001489
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => 2
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000021
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000021: 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] => 1
[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] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[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] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 2
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 2
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 2
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000157
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ 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]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 1
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 2
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000228
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,2] => [2]
=> []
=> 0
[2,1] => [1,1]
=> [1]
=> 1
[1,2,3] => [3]
=> []
=> 0
[1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [1,1,1]
=> [1,1]
=> 2
[1,2,3,4] => [4]
=> []
=> 0
[1,2,4,3] => [3,1]
=> [1]
=> 1
[1,3,2,4] => [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,4,1] => [3,1]
=> [1]
=> 1
[2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 2
[3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 2
[4,1,2,3] => [3,1]
=> [1]
=> 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> []
=> 0
[1,2,3,5,4] => [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [3,2]
=> [2]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000316
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000316: 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] => 1
[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] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[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] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 2
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 2
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 2
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 2
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000325
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000325: 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] => [2,1] => 2 = 1 + 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] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 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] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
Description
The width 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 width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000470: 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] => [2,1] => 2 = 1 + 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] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 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] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000738
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000051The size of the left subtree of a binary tree. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000224The sorting index of a permutation. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000377The dinv defect of an integer partition. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000829The Ulam distance of a permutation to the identity permutation. St000354The number of recoils of a permutation. St000653The last descent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000956The maximal displacement of a permutation. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001626The number of maximal proper sublattices of a lattice. St000454The largest eigenvalue of a graph if it is integral.
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