- FindStat

Your data matches 570 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000267
(load all 22 compositions to match this statistic)

Mapping

St000267: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 1
([],3)
 => 1
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => 1
([(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => 1
([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => 1
([(4,5)],6)
 => 1
([(3,5),(4,5)],6)
 => 1
([(2,5),(3,4)],6)
 => 1
([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => 1
([(5,6)],7)
 => 1
([(4,6),(5,6)],7)
 => 1
([(3,6),(4,5)],7)
 => 1
([(4,5),(4,6),(5,6)],7)
 => 3

Description

The number of maximal spanning forests contained in a graph. A maximal spanning forest in a graph is a maximal acyclic subgraph. In other words, a spanning forest is a union of spanning trees in all connected components. See also [1] for this and further definitions. For connected graphs, this is the same as [[St000096]].

Matching statistic: St001546
(load all 22 compositions to match this statistic)

Mapping

St001546: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 1
([],3)
 => 1
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => 1
([(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => 1
([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => 1
([(4,5)],6)
 => 1
([(3,5),(4,5)],6)
 => 1
([(2,5),(3,4)],6)
 => 1
([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => 1
([(5,6)],7)
 => 1
([(4,6),(5,6)],7)
 => 1
([(3,6),(4,5)],7)
 => 1
([(4,5),(4,6),(5,6)],7)
 => 3

Description

The number of monomials in the Tutte polynomial of a graph.

Matching statistic: St000097
(load all 6 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(3,5),(4,5)],6)
 => ([],4)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => ([],7)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(4,6),(5,6)],7)
 => ([],5)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(5,6)],7)
 => 3

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098
(load all 6 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(3,5),(4,5)],6)
 => ([],4)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => ([],7)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(4,6),(5,6)],7)
 => ([],5)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(5,6)],7)
 => 3

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000172
(load all 8 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(3,5),(4,5)],6)
 => ([],4)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => ([],7)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(4,6),(5,6)],7)
 => ([],5)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(5,6)],7)
 => 3

Description

The Grundy number of a graph. The Grundy number

$\Gamma(G)$ is defined to be the largest

$k$ such that

$G$ admits a greedy

$k$ -coloring. Any order of the vertices of

$G$ induces a greedy coloring by assigning to the

$i$ -th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that

$\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$ , where

$\chi(G)$ is the chromatic number of

$G$ ([[St000098]]), and where

$\Delta(G)$ is the maximal degree of a vertex of

$G$ ([[St000171]]).

Matching statistic: St000271
(load all 7 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
St000271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],6)
 => ([],1)
 => 1
([(4,5)],6)
 => ([(1,2)],3)
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],7)
 => ([],1)
 => 1
([(5,6)],7)
 => ([(1,2)],3)
 => 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 3

Description

The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.

Matching statistic: St000321
(load all 4 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 1
([],2)
 => []
 => 1
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 1
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 1
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 1
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 1
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 1
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 1
([(3,6),(4,5)],7)
 => [1,1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3

Description

The number of integer partitions of n that are dominated by an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition

$\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if

$\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all

$k$ .

Matching statistic: St000345
(load all 5 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 1
([],2)
 => []
 => 1
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 1
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 1
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 1
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 1
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 1
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 1
([(3,6),(4,5)],7)
 => [1,1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

Matching statistic: St000363
(load all 7 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(3,5),(4,5)],6)
 => ([],4)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 3
([],7)
 => ([],7)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(4,6),(5,6)],7)
 => ([],5)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(5,6)],7)
 => 3

Description

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of

$G$ . This is also the number of independent sets of maximal cardinality of

$G$ .

Matching statistic: St000935
(load all 5 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 1
([],2)
 => []
 => 1
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 1
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 1
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 1
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 1
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 1
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 1
([(3,6),(4,5)],7)
 => [1,1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3

Description

The number of ordered refinements of an integer partition. This is, for an integer partition

$\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition

$\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices

$1 = a_0 < \ldots < a_n = m$ with

$\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$ .

The following 560 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001694The number of maximal dissociation sets in a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000346The number of coarsenings of a partition. St000469The distinguishing number of a graph. St000617The number of global maxima of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000008The major index of the composition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000377The dinv defect of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000010The length of the partition. St000096The number of spanning trees of a graph. St000763The sum of the positions of the strong records of an integer composition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000157The number of descents of a standard tableau. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001176The size of a partition minus its first part. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000309The number of vertices with even degree. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001395The number of strictly unfriendly partitions of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000302The determinant of the distance matrix of a connected graph. St000310The minimal degree of a vertex of a graph. St000351The determinant of the adjacency matrix of a graph. St001119The length of a shortest maximal path in a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000315The number of isolated vertices of a graph. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001757The number of orbits of toric promotion on a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001834The number of non-isomorphic minors of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000088The row sums of the character table of the symmetric group. St000184The size of the centralizer of any permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001739The number of graphs with the same edge polytope as the given graph. St001763The Hurwitz number of an integer partition. St001827The number of two-component spanning forests of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. 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. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given 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. St000349The number of different adjacency matrices of a graph. St000378The diagonal inversion number of an integer partition. St000517The Kreweras number of an integer partition. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000667The greatest common divisor of the parts of the partition. St000983The length of the longest alternating subword. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001312Number of parabolic noncrossing partitions indexed by the composition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001386The number of prime labellings of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001924The number of cells in an integer partition whose arm and leg length coincide. St001957The number of Hasse diagrams with a given underlying undirected graph. St000311The number of vertices of odd degree in a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000691The number of changes of a binary word. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000979Half of MacMahon's equal index of a Dyck path. St000995The largest even part of an integer partition. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001522The total irregularity of a graph. St001541The Gini index of an integer partition. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001721The degree of a binary word. St001814The number of partitions interlacing the given partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001118The acyclic chromatic index of a graph. St000005The bounce statistic of a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000120The number of left tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000947The major index east count of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001933The largest multiplicity of a part in an integer partition. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000439The position of the first down step of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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

$eA$ . St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000012The area of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000456The monochromatic index of a connected graph. St000532The total number of rook placements on a Ferrers board. St000655The length of the minimal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000946The sum of the skew hook positions in a Dyck path. 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,. St000984The number of boxes below precisely one peak. 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. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two 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

$g$ . St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001273The projective dimension of the first term in an injective coresolution of 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. 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. St001481The minimal height of a peak of a Dyck path. St001485The modular major index of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001564The value of the forgotten symmetric functions when all variables set to 1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000038The product of the heights of the descending steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. 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. St000784The maximum of the length and the largest part of the integer partition. 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. St000867The sum of the hook lengths in the first row of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001127The sum of the squares of the parts of a partition. St001172The number of 1-rises at odd height of a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001808The box weight or horizontal decoration of a Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000762The sum of the positions of the weak records of an integer composition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000145The Dyson rank of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000567The sum of the products of all pairs of parts. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000438The position of the last up step in a Dyck path. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St000981The length of the longest zigzag subpath. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001098The 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 vertex labelled trees. St000978The sum of the positions of double down-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions 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. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001901The largest multiplicity of an irreducible representation 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St001875The number of simple modules with projective dimension at most 1. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . 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

$eA$ . 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

$eA$ . St000699The toughness times the least common multiple of 1,. St000422The energy of a graph, if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000379The number of Hamiltonian cycles in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. 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. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000914The sum of the values of the Möbius function of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001095The number of non-isomorphic posets with precisely one further covering relation. St000782The indicator function of whether a given perfect matching is an L & P matching.