- FindStat

Your data matches 287 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000093
(load all 12 compositions to match this statistic)

Mapping

St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000097
(load all 14 compositions to match this statistic)

Mapping

St000097: Graphs ⟶ 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
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([],5)
 => 1 = 0 + 1
([(3,4)],5)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3)],5)
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1

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: St000381
(load all 27 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer composition.

Matching statistic: St000382
(load all 6 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St001316
(load all 5 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.

Matching statistic: St001656
(load all 5 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The monophonic position number of a graph. A subset

$M$ of the vertex set of a graph is a monophonic position set if no three vertices of

$M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.

Matching statistic: St001392

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

Matching statistic: St001918

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let

$\lambda$ be an integer partition of

$n$ and let

$N$ be the least common multiple of the parts of

$\lambda$ . Fix an arbitrary permutation

$\pi$ of cycle type

$\lambda$ . Then

$\pi$ induces a cyclic action of order

$N$ on

$\{1,\dots,n\}$ . The corresponding character can be identified with the cyclic sieving polynomial

$C_\lambda(q)$ of this action, modulo

$q^N-1$ . Explicitly, it is

$\sum_{p\in\lambda} [p]_{q^{N/p}},$ where

$[p]_q = 1+\dots+q^{p-1}$ is the

$q$ -integer. This statistic records the degree of

$C_\lambda(q)$ . Equivalently, it equals

$\left(1 - \frac{1}{\lambda_1}\right) N,$ where

$\lambda_1$ is the largest part of

$\lambda$ . The statistic is undefined for the empty partition.

Matching statistic: St000013
(load all 3 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000025
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

The following 277 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000026The position of the first return of a Dyck path. St000147The largest part of an integer partition. St000273The domination number of a graph. St000383The last part of an integer composition. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000917The open packing number of a graph. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001654The monophonic hull number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001949The rigidity index of a graph. St000439The position of the first down step of a Dyck path. St001655The general position number of a graph. St001814The number of partitions interlacing the given partition. St000010The length of the partition. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000392The length of the longest run of ones in a binary word. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001323The independence gap of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001777The number of weak descents in an integer composition. St000148The number of odd parts of a partition. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000363The number of minimal vertex covers of a graph. St000548The number of different non-empty partial sums of an integer partition. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001530The depth of a Dyck path. St001933The largest multiplicity of a part in an integer partition. 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

$c_{n-1}$ . St001674The number of vertices of the largest induced star graph in the graph. St001330The hat guessing number of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001571The Cartan determinant of the integer partition. St000160The multiplicity of the smallest part of a partition. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph 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. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St000479The Ramsey number of a graph. St001645The pebbling number of a connected graph. St000288The number of ones in a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001812The biclique partition number of a graph. St001834The number of non-isomorphic minors of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001093The detour number of a graph. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St000874The position of the last double rise in a Dyck path. St001644The dimension of a graph. St000157The number of descents of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000260The radius of a connected graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001638The book thickness of a graph. St000259The diameter of a connected graph. 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. St001480The number of simple summands of the module J^2/J^3. St000144The pyramid weight of the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001672The restrained domination number of a graph. St001060The distinguishing index of a graph. 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. St000738The first entry in the last row of a standard tableau. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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). St000331The number of upper interactions of 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. 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001118The acyclic chromatic index of a graph. St000993The multiplicity of the largest part of an integer partition. St000474Dyson's crank of a partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001545The second Elser number of a connected graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000015The number of peaks 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. St000120The number of left tunnels of a Dyck path. St000228The size of a partition. St000293The number of inversions of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000459The hook length of the base cell of a partition. 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. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

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

$i>0$ for 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. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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

$g$ . 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: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001488The number of corners of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001614The cyclic permutation representation number of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001884The number of borders of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001271The competition number of a graph. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001108The 2-dynamic chromatic number of a graph. St000422The energy of a graph, if it is integral. St001621The number of atoms of a lattice. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000675The number of centered multitunnels of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice. 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

$eA$ . St001877Number of indecomposable injective modules with projective dimension 2. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of 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. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the 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. 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. 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. St000776The maximal multiplicity of an eigenvalue in a graph.