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Your data matches 580 different statistics following compositions of up to 3 maps.
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Matching statistic: St000805
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000805: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 1
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 1
[2,3] => 1
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St000807
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,1] => 0 = 1 - 1
[2] => 0 = 1 - 1
[1,1,1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[3] => 0 = 1 - 1
[1,1,1,1] => 0 = 1 - 1
[1,1,2] => 0 = 1 - 1
[1,2,1] => 0 = 1 - 1
[1,3] => 0 = 1 - 1
[2,1,1] => 0 = 1 - 1
[2,2] => 0 = 1 - 1
[3,1] => 0 = 1 - 1
[4] => 0 = 1 - 1
[1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => 0 = 1 - 1
[1,1,2,1] => 0 = 1 - 1
[1,1,3] => 0 = 1 - 1
[1,2,1,1] => 0 = 1 - 1
[1,2,2] => 0 = 1 - 1
[1,3,1] => 0 = 1 - 1
[1,4] => 0 = 1 - 1
[2,1,1,1] => 0 = 1 - 1
[2,1,2] => 1 = 2 - 1
[2,2,1] => 0 = 1 - 1
[2,3] => 0 = 1 - 1
[3,1,1] => 0 = 1 - 1
[3,2] => 0 = 1 - 1
[4,1] => 0 = 1 - 1
[5] => 0 = 1 - 1
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000816
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 1
[2] => [1] => 1
[1,1,1] => [3] => 1
[1,2] => [1,1] => 1
[2,1] => [1,1] => 1
[3] => [1] => 1
[1,1,1,1] => [4] => 1
[1,1,2] => [2,1] => 1
[1,2,1] => [1,1,1] => 1
[1,3] => [1,1] => 1
[2,1,1] => [1,2] => 1
[2,2] => [2] => 1
[3,1] => [1,1] => 1
[4] => [1] => 1
[1,1,1,1,1] => [5] => 1
[1,1,1,2] => [3,1] => 1
[1,1,2,1] => [2,1,1] => 1
[1,1,3] => [2,1] => 1
[1,2,1,1] => [1,1,2] => 1
[1,2,2] => [1,2] => 1
[1,3,1] => [1,1,1] => 1
[1,4] => [1,1] => 1
[2,1,1,1] => [1,3] => 2
[2,1,2] => [1,1,1] => 1
[2,2,1] => [2,1] => 1
[2,3] => [1,1] => 1
[3,1,1] => [1,2] => 1
[3,2] => [1,1] => 1
[4,1] => [1,1] => 1
[5] => [1] => 1
Description
The number of standard composition tableaux of the composition.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
Matching statistic: St001256
(load all 176 compositions to match this statistic)
(load all 176 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001256: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St001518
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> 1
[2] => ([],2)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> 1
[3] => ([],3)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4] => ([],4)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5] => ([],5)
=> 1
Description
The number of graphs with the same ordinary spectrum as the given graph.
Matching statistic: St000322
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0 = 1 - 1
[1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[2] => ([],2)
=> 0 = 1 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3] => ([],3)
=> 0 = 1 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4] => ([],4)
=> 0 = 1 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5] => ([],5)
=> 0 = 1 - 1
Description
The skewness of a graph.
For a graph $G$, the '''skewness''' of $G$ is the minimum number of edges of $G$ whose removal results in a planar graph.
Matching statistic: St000323
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000323: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000323: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0 = 1 - 1
[1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[2] => ([],2)
=> 0 = 1 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3] => ([],3)
=> 0 = 1 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4] => ([],4)
=> 0 = 1 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5] => ([],5)
=> 0 = 1 - 1
Description
The minimal crossing number of a graph.
A '''drawing''' of a graph $G$ is a drawing in $\mathbb{R}^2$ such that
* the vertices of $G$ are distinct points,
* the edges of $G$ are simple curves joining their endpoints,
* no edge passes through a vertex, and
* no three edges cross in a common point.
The '''minimal crossing number''' of $G$ is then the minimal number of crossings of edges in a drawing of $G$.
In particular, a graph is planar if and only if its minimal crossing number is $0$.
It is moreover conjectured that the crossing number of the complete graph $K_n$ [1] is
$$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$$
and the crossing number of the complete bipartite graph $K_{n,m}$ [2] is
$$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$$
A general algorithm to compute the crossing number is e.g. given in [3].
This statistics data was provided by Markus Chimani [6].
Matching statistic: St000370
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000370: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0 = 1 - 1
[1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[2] => ([],2)
=> 0 = 1 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3] => ([],3)
=> 0 = 1 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4] => ([],4)
=> 0 = 1 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5] => ([],5)
=> 0 = 1 - 1
Description
The genus of a graph.
This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.
Matching statistic: St000047
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,1] => [2] => [1] => 1
[2] => [1] => [1] => 1
[1,1,1] => [3] => [1] => 1
[1,2] => [1,1] => [2] => 1
[2,1] => [1,1] => [2] => 1
[3] => [1] => [1] => 1
[1,1,1,1] => [4] => [1] => 1
[1,1,2] => [2,1] => [1,1] => 1
[1,2,1] => [1,1,1] => [3] => 1
[1,3] => [1,1] => [2] => 1
[2,1,1] => [1,2] => [1,1] => 1
[2,2] => [2] => [1] => 1
[3,1] => [1,1] => [2] => 1
[4] => [1] => [1] => 1
[1,1,1,1,1] => [5] => [1] => 1
[1,1,1,2] => [3,1] => [1,1] => 1
[1,1,2,1] => [2,1,1] => [1,2] => 1
[1,1,3] => [2,1] => [1,1] => 1
[1,2,1,1] => [1,1,2] => [2,1] => 2
[1,2,2] => [1,2] => [1,1] => 1
[1,3,1] => [1,1,1] => [3] => 1
[1,4] => [1,1] => [2] => 1
[2,1,1,1] => [1,3] => [1,1] => 1
[2,1,2] => [1,1,1] => [3] => 1
[2,2,1] => [2,1] => [1,1] => 1
[2,3] => [1,1] => [2] => 1
[3,1,1] => [1,2] => [1,1] => 1
[3,2] => [1,1] => [2] => 1
[4,1] => [1,1] => [2] => 1
[5] => [1] => [1] => 1
Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000079
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000079: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000079: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
Description
The number of alternating sign matrices for a given Dyck path.
The Dyck path is given by the last diagonal of the monotone triangle corresponding to an alternating sign matrix.
The following 570 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000390The number of runs of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000920The logarithmic height of a Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra 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. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000660The number of rises of length at least 3 of a Dyck path. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001715The number of non-records in a permutation. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001871The number of triconnected components of a graph. St000003The number of standard Young tableaux of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000048The multinomial of the parts of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000277The number of ribbon shaped standard tableaux. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000767The number of runs in an integer composition. 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. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. 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. St001487The number of inner corners of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001716The 1-improper chromatic number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000089The absolute variation of a composition. St000090The variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. 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. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000223The number of nestings in the permutation. St000252The number of nodes of degree 3 of a binary tree. 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. St000290The major index of a binary word. St000292The number of ascents of a binary word. St000313The number of degree 2 vertices of a graph. St000317The cycle descent number of a permutation. St000348The non-inversion sum of a binary word. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun 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 $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. 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. St000516The number of stretching pairs of a permutation. St000552The number of cut vertices of a graph. St000623The number of occurrences of the pattern 52341 in a permutation. St000629The defect of a binary word. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000731The number of double exceedences of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three 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. 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. St001214The aft of an integer partition. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001350Half of the Albertson index of a graph. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. 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. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000962The 3-shifted major index of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001162The minimum jump of a permutation. St001220The width of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000842The breadth of a permutation. St000872The number of very big descents of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001703The villainy of a graph. St001712The number of natural descents of a standard Young tableau. St001731The factorization defect of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000488The number of cycles of a permutation of length at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001204Call 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. 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. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000219The number of occurrences of the pattern 231 in a permutation. St000914The sum of the values of the Möbius function of a poset. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000284The Plancherel distribution on integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. 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. St000886The number of permutations with the same antidiagonal sums. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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. St001128The exponens consonantiae of a partition. St001129The product of the squares of the parts of a partition. 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 $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001389The number of partitions of the same length below the given integer partition. St001481The minimal height of a peak of a Dyck path. St001490The number of connected components of a skew partition. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001597The Frobenius rank of a skew partition. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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. 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. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001496The number of graphs with the same Laplacian spectrum as the given graph. 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. St000287The number of connected components of a graph. St001765The number of connected components of the friends and strangers graph. St000286The number of connected components of the complement of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001881The number of factors of a lattice as a Cartesian product of lattices. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001964The interval resolution global dimension of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000297The number of leading ones in a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000068The number of minimal elements in a poset. St000053The number of valleys of the Dyck path. St000260The radius of a connected graph. St000306The bounce count of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001884The number of borders of a binary word. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000657The smallest part of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001260The permanent of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000894The trace of an alternating sign matrix. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001429The number of negative entries in a signed permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000100The number of linear extensions of a poset. 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. St000635The number of strictly order preserving maps of a poset into itself. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000456The monochromatic index of a connected graph. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000039The number of crossings of a permutation. St000058The order of a permutation. St000084The number of subtrees. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000295The length of the border of a binary word. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001645The pebbling number of a connected graph. St000259The diameter 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.
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