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Your data matches 663 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000496
(load all 13 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ 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] => {{1,2}}
 => 0
[1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => {{1},{2,3}}
 => 0
[2,1,3] => {{1,2},{3}}
 => 0
[2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => {{1,2,3}}
 => 0
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => 0
[1,3,2,4] => {{1},{2,3},{4}}
 => 0
[1,3,4,2] => {{1},{2,3,4}}
 => 0
[1,4,2,3] => {{1},{2,3,4}}
 => 0
[1,4,3,2] => {{1},{2,4},{3}}
 => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => 0
[2,1,4,3] => {{1,2},{3,4}}
 => 0
[2,3,1,4] => {{1,2,3},{4}}
 => 0
[2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => {{1,2,3,4}}
 => 0
[3,1,2,4] => {{1,2,3},{4}}
 => 0
[3,1,4,2] => {{1,2,3,4}}
 => 0
[3,2,1,4] => {{1,3},{2},{4}}
 => 1
[4,1,2,3] => {{1,2,3,4}}
 => 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 0
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 0
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 0
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 0
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 1
[1,5,2,3,4] => {{1},{2,3,4,5}}
 => 0
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 0
[2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 0
[2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 0
[2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0
[2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0
[2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 0
[2,3,1,5,4] => {{1,2,3},{4,5}}
 => 0
[2,3,4,1,5] => {{1,2,3,4},{5}}
 => 0
[2,4,1,3,5] => {{1,2,3,4},{5}}
 => 0
[3,1,2,4,5] => {{1,2,3},{4},{5}}
 => 0
[3,1,2,5,4] => {{1,2,3},{4,5}}
 => 0

Description

The rcs statistic of a set partition. Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ . According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of

$S$ is given by a pair

$i > j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a < b$ .

Matching statistic: St000752
(load all 14 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 0
[1,2] => [1,1]
 => 0
[2,1] => [2]
 => 0
[1,2,3] => [1,1,1]
 => 0
[1,3,2] => [2,1]
 => 0
[2,1,3] => [2,1]
 => 0
[2,3,1] => [2,1]
 => 0
[3,1,2] => [2,1]
 => 0
[3,2,1] => [3]
 => 1
[1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => 0
[1,3,4,2] => [2,1,1]
 => 0
[1,4,2,3] => [2,1,1]
 => 0
[1,4,3,2] => [3,1]
 => 1
[2,1,3,4] => [2,1,1]
 => 0
[2,1,4,3] => [2,2]
 => 0
[2,3,1,4] => [2,1,1]
 => 0
[2,3,4,1] => [2,1,1]
 => 0
[2,4,1,3] => [2,1,1]
 => 0
[3,1,2,4] => [2,1,1]
 => 0
[3,1,4,2] => [2,2]
 => 0
[3,2,1,4] => [3,1]
 => 1
[4,1,2,3] => [2,1,1]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => 0
[1,2,4,5,3] => [2,1,1,1]
 => 0
[1,2,5,3,4] => [2,1,1,1]
 => 0
[1,2,5,4,3] => [3,1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => 0
[1,3,4,2,5] => [2,1,1,1]
 => 0
[1,3,4,5,2] => [2,1,1,1]
 => 0
[1,3,5,2,4] => [2,1,1,1]
 => 0
[1,4,2,3,5] => [2,1,1,1]
 => 0
[1,4,2,5,3] => [2,2,1]
 => 0
[1,4,3,2,5] => [3,1,1]
 => 1
[1,5,2,3,4] => [2,1,1,1]
 => 0
[2,1,3,4,5] => [2,1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => 0
[2,1,4,3,5] => [2,2,1]
 => 0
[2,1,4,5,3] => [2,2,1]
 => 0
[2,1,5,3,4] => [2,2,1]
 => 0
[2,3,1,4,5] => [2,1,1,1]
 => 0
[2,3,1,5,4] => [2,2,1]
 => 0
[2,3,4,1,5] => [2,1,1,1]
 => 0
[2,4,1,3,5] => [2,1,1,1]
 => 0
[3,1,2,4,5] => [2,1,1,1]
 => 0
[3,1,2,5,4] => [2,2,1]
 => 0

Description

The Grundy value for the game 'Couples are forever' on an integer partition. Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.

Matching statistic: St001311
(load all 11 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001311: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as

$c + m - n$ , where

$c$ is the number of connected components,

$m$ is the number of edges and

$n$ is the number of vertices.

Matching statistic: St001317
(load all 11 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001317: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices

$a < b < c$ such that

$(a,c)$ and

$(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001328
(load all 12 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices

$a < b < c$ such that

$(a,b)$ and

$(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001331
(load all 12 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001331: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.

Matching statistic: St001335
(load all 12 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The cardinality of a minimal cycle-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains all cycles.

Matching statistic: St001336
(load all 16 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The minimal number of vertices in a graph whose complement is triangle-free.

Matching statistic: St001736
(load all 11 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001736: 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)
 => 0
[1,2,3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0

Description

The total number of cycles in a graph. For example, the complete graph on four vertices has four triangles and three different four-cycles.

Matching statistic: St001781
(load all 11 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St001781: Set partitions ⟶ 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] => {{1,2}}
 => 0
[1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => {{1},{2,3}}
 => 0
[2,1,3] => {{1,2},{3}}
 => 0
[2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => {{1,2,3}}
 => 0
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => 0
[1,3,2,4] => {{1},{2,3},{4}}
 => 0
[1,3,4,2] => {{1},{2,3,4}}
 => 0
[1,4,2,3] => {{1},{2,3,4}}
 => 0
[1,4,3,2] => {{1},{2,4},{3}}
 => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => 0
[2,1,4,3] => {{1,2},{3,4}}
 => 0
[2,3,1,4] => {{1,2,3},{4}}
 => 0
[2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => {{1,2,3,4}}
 => 0
[3,1,2,4] => {{1,2,3},{4}}
 => 0
[3,1,4,2] => {{1,2,3,4}}
 => 0
[3,2,1,4] => {{1,3},{2},{4}}
 => 1
[4,1,2,3] => {{1,2,3,4}}
 => 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 0
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 0
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 0
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 0
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 1
[1,5,2,3,4] => {{1},{2,3,4,5}}
 => 0
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 0
[2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 0
[2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 0
[2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0
[2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0
[2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 0
[2,3,1,5,4] => {{1,2,3},{4,5}}
 => 0
[2,3,4,1,5] => {{1,2,3,4},{5}}
 => 0
[2,4,1,3,5] => {{1,2,3,4},{5}}
 => 0
[3,1,2,4,5] => {{1,2,3},{4},{5}}
 => 0
[3,1,2,5,4] => {{1,2,3},{4,5}}
 => 0

Description

The interlacing number of a set partition. Let

$\pi$ be a set partition of

$\{1,\dots,n\}$ with

$k$ blocks. To each block of

$\pi$ we add the element

$\infty$ , which is larger than

$n$ . Then, an interlacing of

$\pi$ is a pair of blocks

$B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and

$C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index

$1\leq i\leq \min(b, c)$ , such that

$B_i < C_i < B_{i+1} < C_{i+1}$ .

The following 653 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001797The number of overfull subgraphs of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001512The minimum rank of a graph. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000048The multinomial of the parts of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000346The number of coarsenings of a partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000883The number of longest increasing subsequences of a permutation. St000920The logarithmic height of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001261The Castelnuovo-Mumford regularity of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000002The number of occurrences of the pattern 123 in a permutation. St000024The number of double up and double down steps of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000091The descent variation of a composition. 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. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000169The cocharge of a standard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000223The number of nestings in the permutation. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000463The number of admissible inversions of a permutation. St000552The number of cut vertices of a graph. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000761The number of ascents in an integer composition. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001252Half the sum of the even parts of a partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. 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. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001928The number of non-overlapping descents in a permutation. St000040The number of regions of the inversion arrangement of a permutation. St000058The order of a permutation. St000201The number of leaf nodes in a binary tree. St000292The number of ascents of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000679The pruning number of an ordered tree. St000733The row containing the largest entry of a standard tableau. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000862The number of parts of the shifted shape of a permutation. St000877The depth of the binary word interpreted as a path. St000935The number of ordered refinements of an integer partition. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St000290The major index of a binary word. St000291The number of descents of a binary word. St000354The number of recoils of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000565The major index of a set partition. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St000485The length of the longest cycle of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. 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. St000035The number of left outer peaks of a permutation. St000095The number of triangles of a graph. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St000110The number of permutations less than or equal to a permutation in left weak order. St001718The number of non-empty open intervals in a poset. St000065The number of entries equal to -1 in an alternating sign matrix. St000360The number of occurrences of the pattern 32-1. St001411The number of patterns 321 or 3412 in a permutation. St001728The number of invisible descents of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001596The number of two-by-two squares inside a skew partition. St001597The Frobenius rank of a skew partition. St000299The number of nonisomorphic vertex-induced subtrees. St001642The Prague dimension of a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001933The largest multiplicity of a part in an integer partition. St000010The length of the partition. St000012The area of a Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000228The size of a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000992The alternating sum of the parts of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001127The sum of the squares of the parts of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001484The number of singletons of an integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. 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. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. 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. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St000294The number of distinct factors of a binary word. St000439The position of the first down step of a Dyck path. St000518The number of distinct subsequences in a binary word. 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. 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. St000806The semiperimeter of the associated bargraph. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima 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. St001114The number of odd descents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000990The first ascent of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000647The number of big descents of a permutation. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000358The number of occurrences of the pattern 31-2. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000886The number of permutations with the same antidiagonal sums. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000934The 2-degree of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001568The smallest positive integer that does not appear twice in the partition. St000039The number of crossings of a permutation. St000649The number of 3-excedences of a permutation. St000663The number of right floats of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001731The factorization defect of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St000100The number of linear extensions 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. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. 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. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001396Number of triples of incomparable elements in a finite poset. St000367The number of simsun double descents of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000264The girth of a graph, which is not a tree. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000218The number of occurrences of the pattern 213 in a permutation. St000353The number of inner valleys of a permutation. St000534The number of 2-rises of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000255The number of reduced Kogan faces with the permutation as type. St000365The number of double ascents of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000217The number of occurrences of the pattern 312 in a permutation. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000948The chromatic discriminant of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000089The absolute variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001673The degree of asymmetry of an integer composition. St001871The number of triconnected components of a graph. St001874Lusztig's a-function for the symmetric group. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000047The number of standard immaculate tableaux of a given shape. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000277The number of ribbon shaped standard tableaux. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000889The number of alternating sign matrices with the same antidiagonal sums. St000903The number of different parts of an integer composition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001758The number of orbits of promotion on a graph. St001917The order of toric promotion on the set of labellings of a graph. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001082The number of boxed occurrences of 123 in a permutation. St000216The absolute length of a permutation. St000646The number of big ascents of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000837The number of ascents of distance 2 of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001859The number of factors of the Stanley symmetric function associated with a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001531Number of partial orders contained in the poset determined by the Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001959The product of the heights of the peaks of a Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000477The weight of a partition according to Alladi. St000937The number of positive values of the symmetric group character corresponding to the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001556The number of inversions of the third entry of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001586The number of odd parts smaller than the largest even part in an integer partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001964The interval resolution global dimension of a poset. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001513The number of nested exceedences of a permutation. St001684The reduced word complexity of a permutation. 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. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St000456The monochromatic index of a connected graph. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001770The number of facets of a certain subword complex associated with the signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000741The Colin de Verdière graph invariant. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .