- FindStat

Your data matches 475 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001725
(load all 6 compositions to match this statistic)

Mapping

St001725: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.

Matching statistic: St000987
(load all 6 compositions to match this statistic)

Mapping

St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

Matching statistic: St000345
(load all 4 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 1
([],2)
 => [1,1]
 => 1
([(0,1)],2)
 => [2]
 => 2
([],3)
 => [1,1,1]
 => 1
([(1,2)],3)
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => [1,1,1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => 2
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 3
([(1,4),(2,3)],5)
 => [2,2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([],6)
 => [1,1,1,1,1,1]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => 2
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 3
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3
([],7)
 => [1,1,1,1,1,1,1]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => 2
([(4,6),(5,6)],7)
 => [2,2,1,1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => 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: St000935
(load all 4 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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

Matching statistic: St001389
(load all 9 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St000362
(load all 3 compositions to match this statistic)

Mapping

Mp00147: Graphs —square⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

Matching statistic: St000469

Mapping

Mp00147: Graphs —square⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000469: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The distinguishing number of a graph. This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring. For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].

Matching statistic: St000507
(load all 4 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => [[1,2],[3,4]]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 3
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 3
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 2
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 2
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 3
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 3
([],7)
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 2
([(4,6),(5,6)],7)
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 3
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 3

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000528

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000912

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of maximal antichains in a poset.

The following 465 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000008The major index of the composition. St000080The rank of the poset. St000171The degree of the graph. St000377The dinv defect of an integer partition. St000778The metric dimension of a graph. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001345The Hamming dimension of a graph. St001782The order of rowmotion on the set of order ideals of a poset. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St000026The position of the first return of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000532The total number of rook placements on a Ferrers board. St000722The number of different neighbourhoods in a graph. St000738The first entry in the last row of a standard tableau. St000839The largest opener of a set partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000074The number of special entries. St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000228The size of a partition. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000459The hook length of the base cell of a partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St000676The number of odd rises of a Dyck path. St000784The maximum of the length and the largest part of the integer partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. 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. St001759The Rajchgot index of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000054The first entry of the permutation. 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. St000141The maximum drop size of a permutation. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000653The last descent of a permutation. St000794The mak of a permutation. St000240The number of indices that are not small excedances. St000740The last entry of a permutation. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001497The position of the largest weak excedence of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000387The matching number of a graph. St000638The number of up-down runs of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000004The major index of a permutation. St000051The size of the left subtree of a binary tree. St000089The absolute variation of a composition. St000316The number of non-left-to-right-maxima of a permutation. St000833The comajor index of a permutation. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001480The number of simple summands of the module J^2/J^3. St000258The burning number of a graph. St000918The 2-limited packing 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001340The cardinality of a minimal non-edge isolating set of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. 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. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St000010The length of the partition. St000184The size of the centralizer of any permutation of given cycle type. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000643The size of the largest orbit of antichains under Panyushev complementation. St000644The number of graphs with given frequency partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001486The number of corners of the ribbon associated with an integer composition. St001488The number of corners of a skew partition. 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). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St001746The coalition number of a graph. St000149The number of cells of the partition whose leg is zero and arm is odd. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001031The height of the bicoloured Motzkin path associated with the 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. 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). St001956The comajor index for set-valued two-row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000439The position of the first down step of a Dyck path. St000626The minimal period of a binary word. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. 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

$c_{n-1}$ . St000981The length of the longest zigzag subpath. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001437The flex of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000015The number of peaks of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000443The number of long tunnels of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000922The minimal number such that all substrings of this length are unique. St000946The sum of the skew hook positions in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001471The magnitude of a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001607The number of coloured graphs 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. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a 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. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001955The number of natural descents for set-valued two row standard Young tableaux. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000053The number of valleys of the Dyck path. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000376The bounce deficit of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000628The balance of a binary word. St000674The number of hills 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001011Number of simple modules of projective dimension 2 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra