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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000829
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
St000829: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St000354
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
[1,4,5,3,2] => [1,2,5,4,3] => 2
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
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
[1,2] => ([],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
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],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
[1,2] => [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
[1,4,5,3,2] => [3,1,1]
=> 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
[1,2] => [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
[1,4,5,3,2] => [1,2,5,4,3] => 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: 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
[1,2] => [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
[1,4,5,3,2] => [1,2,5,4,3] => [[1,2,3],[4],[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
[1,2] => [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
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
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
[1,2] => [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
[1,4,5,3,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 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]].
Matching statistic: St000024
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000141
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,2,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,5,2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,4,5,3,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000211The rank of the set partition. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000377The dinv defect of an integer partition. St000662The staircase size of the code of a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001726The number of visible inversions of a permutation. St000054The first entry of the permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the 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. 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. St000703The number of deficiencies of a permutation. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000245The number of ascents of a permutation. St000653The last descent of a permutation. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000051The size of the left subtree of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a 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. St000740The last entry of a permutation. 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. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice. St000454The largest eigenvalue of a graph if it is integral.
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