- FindStat

Your data matches 462 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000080

Mapping

St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank of the poset.

Matching statistic: St000632
(load all 3 compositions to match this statistic)

Mapping

St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The jump number of the poset. A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St000845
(load all 2 compositions to match this statistic)

Mapping

St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of elements covered by an element in a poset.

Matching statistic: St000846
(load all 2 compositions to match this statistic)

Mapping

St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of elements covering an element of a poset.

Matching statistic: St000528
(load all 2 compositions to match this statistic)

Mapping

St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1 = 0 + 1
([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => 2 = 1 + 1
([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2)],3)
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => 2 = 1 + 1
([(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St001343
(load all 2 compositions to match this statistic)

Mapping

St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1 = 0 + 1
([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => 2 = 1 + 1
([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2)],3)
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => 2 = 1 + 1
([(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1

Description

The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.

Matching statistic: St000272
(load all 7 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000319

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St000536
(load all 7 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The pathwidth of a graph.

The following 452 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000482The (zero)-forcing number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St000778The metric dimension of a graph. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001512The minimum rank of a graph. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000295The length of the border of a binary word. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000741The Colin de Verdière graph invariant. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001727The number of invisible inversions of a permutation. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000005The bounce statistic of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000141The maximum drop size of a permutation. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000653The last descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000833The comajor index of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000876The number of factors in the Catalan decomposition of 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. St000926The clique-coclique number of a graph. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 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. St001110The 3-dynamic chromatic number of a graph. 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center 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. St001415The length of the longest palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. 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. St001725The harmonious chromatic number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000054The first entry of the permutation. St000144The pyramid weight of the Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000839The largest opener of a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001497The position of the largest weak excedence of a permutation. 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. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000473The number of parts of a partition that are strictly bigger than the number of ones. St001280The number of parts of an integer partition that are at least two. St000668The least common multiple of the parts of the partition. St000083The number of left oriented leafs of a binary tree except the first one. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001062The maximal size of a block of a set partition. St001432The order dimension of the partition. St000143The largest repeated part of a partition. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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: St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001720The minimal length of a chain of small intervals in a lattice. St000776The maximal multiplicity of an eigenvalue in a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . 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. 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. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module 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

$eA$ . 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001340The cardinality of a minimal non-edge isolating set of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001933The largest multiplicity of a part in an integer partition. St000015The number of peaks of a Dyck path. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001488The number of corners of a skew 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001568The smallest positive integer that does not appear twice in the partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000271The chromatic index of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000537The cutwidth of a graph. St000552The number of cut vertices of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001393The induced matching number of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000544The cop number of a graph. St000553The number of blocks of a graph. St000785The number of distinct colouring schemes of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001877Number of indecomposable injective modules with projective dimension 2. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000460The hook length of the last cell along the main diagonal of an integer 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000907The number of maximal antichains of minimal length in a poset. 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. St000929The constant term of the character polynomial of an integer 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000068The number of minimal elements in a poset. St000100The number of linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000189The number of elements in the poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000307The number of rowmotion orbits of a poset. 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. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000914The sum of the values of the Möbius function of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001717The largest size of an interval in a poset. St001779The order of promotion on the set of linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000478Another weight of a partition according to Alladi. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001742The difference of the maximal and the minimal degree in a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000450The number of edges minus the number of vertices plus 2 of a graph. St001734The lettericity of a graph. St000315The number of isolated vertices of a graph. St001642The Prague dimension of a graph. St000327The number of cover relations in a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000284The Plancherel distribution on integer partitions. St000420The number of Dyck paths that are weakly above a Dyck path. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000744The length of the path to the largest entry in a standard Young tableau. St000901The cube of the number of standard Young tableaux with shape given by 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. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. 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. St001808The box weight or horizontal decoration of a Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000145The Dyson rank of a partition. St001060The distinguishing index of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. 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. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. 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

$eA$ . St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001545The second Elser number 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.