- FindStat

Your data matches 173 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000147
(load all 8 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 1
[1,2] => [2]
 => 2
[2,1] => [1,1]
 => 1
[1,2,3] => [3]
 => 3
[1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1]
 => 2
[2,3,1] => [2,1]
 => 2
[3,1,2] => [2,1]
 => 2
[3,2,1] => [1,1,1]
 => 1
[1,2,3,4] => [4]
 => 4
[1,2,4,3] => [3,1]
 => 3
[1,3,2,4] => [3,1]
 => 3
[1,3,4,2] => [3,1]
 => 3
[1,4,2,3] => [3,1]
 => 3
[1,4,3,2] => [2,1,1]
 => 2
[2,1,3,4] => [3,1]
 => 3
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => 3
[2,3,4,1] => [3,1]
 => 3
[2,4,1,3] => [2,2]
 => 2
[2,4,3,1] => [2,1,1]
 => 2
[3,1,2,4] => [3,1]
 => 3
[3,1,4,2] => [2,2]
 => 2
[3,2,1,4] => [2,1,1]
 => 2
[3,2,4,1] => [2,1,1]
 => 2
[3,4,1,2] => [2,2]
 => 2
[3,4,2,1] => [2,1,1]
 => 2
[4,1,2,3] => [3,1]
 => 3
[4,1,3,2] => [2,1,1]
 => 2
[4,2,1,3] => [2,1,1]
 => 2
[4,2,3,1] => [2,1,1]
 => 2
[4,3,1,2] => [2,1,1]
 => 2
[4,3,2,1] => [1,1,1,1]
 => 1
[1,2,3,4,5] => [5]
 => 5
[1,2,3,5,4] => [4,1]
 => 4
[1,2,4,3,5] => [4,1]
 => 4
[1,2,4,5,3] => [4,1]
 => 4
[1,2,5,3,4] => [4,1]
 => 4
[1,2,5,4,3] => [3,1,1]
 => 3
[1,3,2,4,5] => [4,1]
 => 4
[1,3,2,5,4] => [3,2]
 => 3
[1,3,4,2,5] => [4,1]
 => 4
[1,3,4,5,2] => [4,1]
 => 4
[1,3,5,2,4] => [3,2]
 => 3
[1,3,5,4,2] => [3,1,1]
 => 3
[1,4,2,3,5] => [4,1]
 => 4
[1,4,2,5,3] => [3,2]
 => 3
[1,4,3,2,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => 3
[1,4,5,2,3] => [3,2]
 => 3

Description

The largest part of an integer partition.

Matching statistic: St001337
(load all 11 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of

$G$ . The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of

$K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].

Matching statistic: St001338
(load all 11 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The upper irredundance number of a graph. A set

$S$ of vertices is irredundant, if there is no vertex in

$S$ , whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of

$S$ . The upper irredundance number is the largest size of a maximal irredundant set. The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of

$K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].

Matching statistic: St000010
(load all 5 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [1]
 => 1
[1,2] => [2]
 => [1,1]
 => 2
[2,1] => [1,1]
 => [2]
 => 1
[1,2,3] => [3]
 => [1,1,1]
 => 3
[1,3,2] => [2,1]
 => [2,1]
 => 2
[2,1,3] => [2,1]
 => [2,1]
 => 2
[2,3,1] => [2,1]
 => [2,1]
 => 2
[3,1,2] => [2,1]
 => [2,1]
 => 2
[3,2,1] => [1,1,1]
 => [3]
 => 1
[1,2,3,4] => [4]
 => [1,1,1,1]
 => 4
[1,2,4,3] => [3,1]
 => [2,1,1]
 => 3
[1,3,2,4] => [3,1]
 => [2,1,1]
 => 3
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 3
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 3
[1,4,3,2] => [2,1,1]
 => [3,1]
 => 2
[2,1,3,4] => [3,1]
 => [2,1,1]
 => 3
[2,1,4,3] => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 3
[2,3,4,1] => [3,1]
 => [2,1,1]
 => 3
[2,4,1,3] => [2,2]
 => [2,2]
 => 2
[2,4,3,1] => [2,1,1]
 => [3,1]
 => 2
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 3
[3,1,4,2] => [2,2]
 => [2,2]
 => 2
[3,2,1,4] => [2,1,1]
 => [3,1]
 => 2
[3,2,4,1] => [2,1,1]
 => [3,1]
 => 2
[3,4,1,2] => [2,2]
 => [2,2]
 => 2
[3,4,2,1] => [2,1,1]
 => [3,1]
 => 2
[4,1,2,3] => [3,1]
 => [2,1,1]
 => 3
[4,1,3,2] => [2,1,1]
 => [3,1]
 => 2
[4,2,1,3] => [2,1,1]
 => [3,1]
 => 2
[4,2,3,1] => [2,1,1]
 => [3,1]
 => 2
[4,3,1,2] => [2,1,1]
 => [3,1]
 => 2
[4,3,2,1] => [1,1,1,1]
 => [4]
 => 1
[1,2,3,4,5] => [5]
 => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => [4,1]
 => [2,1,1,1]
 => 4
[1,2,4,3,5] => [4,1]
 => [2,1,1,1]
 => 4
[1,2,4,5,3] => [4,1]
 => [2,1,1,1]
 => 4
[1,2,5,3,4] => [4,1]
 => [2,1,1,1]
 => 4
[1,2,5,4,3] => [3,1,1]
 => [3,1,1]
 => 3
[1,3,2,4,5] => [4,1]
 => [2,1,1,1]
 => 4
[1,3,2,5,4] => [3,2]
 => [2,2,1]
 => 3
[1,3,4,2,5] => [4,1]
 => [2,1,1,1]
 => 4
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 4
[1,3,5,2,4] => [3,2]
 => [2,2,1]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,2,3,5] => [4,1]
 => [2,1,1,1]
 => 4
[1,4,2,5,3] => [3,2]
 => [2,2,1]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,5,2,3] => [3,2]
 => [2,2,1]
 => 3

Description

The length of the partition.

Matching statistic: St000381
(load all 6 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
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] => 1
[1,2] => ([],2)
 => [2] => 2
[2,1] => ([(0,1)],2)
 => [1,1] => 1
[1,2,3] => ([],3)
 => [3] => 3
[1,3,2] => ([(1,2)],3)
 => [2,1] => 2
[2,1,3] => ([(1,2)],3)
 => [2,1] => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,1] => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,1] => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 1
[1,2,3,4] => ([],4)
 => [4] => 4
[1,2,4,3] => ([(2,3)],4)
 => [3,1] => 3
[1,3,2,4] => ([(2,3)],4)
 => [3,1] => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[2,1,3,4] => ([(2,3)],4)
 => [3,1] => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
[1,2,3,4,5] => ([],5)
 => [5] => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [4,1] => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [4,1] => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [4,1] => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [3,2] => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1] => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 3

Description

The largest part of an integer composition.

Matching statistic: St000382

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
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] => 1
[1,2] => ([],2)
 => [2] => 2
[2,1] => ([(0,1)],2)
 => [1,1] => 1
[1,2,3] => ([],3)
 => [3] => 3
[1,3,2] => ([(1,2)],3)
 => [2,1] => 2
[2,1,3] => ([(1,2)],3)
 => [2,1] => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,1] => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,1] => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 1
[1,2,3,4] => ([],4)
 => [4] => 4
[1,2,4,3] => ([(2,3)],4)
 => [3,1] => 3
[1,3,2,4] => ([(2,3)],4)
 => [3,1] => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[2,1,3,4] => ([(2,3)],4)
 => [3,1] => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
[1,2,3,4,5] => ([],5)
 => [5] => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [4,1] => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [4,1] => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [4,1] => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [3,2] => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1] => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [4,1] => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 3

Description

The first part of an integer composition.

Matching statistic: St000507
(load all 5 compositions to match this statistic)

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion 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
[1,2] => [1,2] => [[1,2]]
 => 2
[2,1] => [2,1] => [[1],[2]]
 => 1
[1,2,3] => [1,2,3] => [[1,2,3]]
 => 3
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => 2
[2,3,1] => [1,3,2] => [[1,2],[3]]
 => 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 3
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => 3
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 3
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 3
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => 3
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => 3
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 4
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 4
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 4
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 3
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => 4
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 3
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 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: St000676
(load all 3 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000734
(load all 2 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [[1]]
 => 1
[1,2] => [2]
 => [[1,2]]
 => 2
[2,1] => [1,1]
 => [[1],[2]]
 => 1
[1,2,3] => [3]
 => [[1,2,3]]
 => 3
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => 2
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => 2
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 4
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => 2
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 5
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 3

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000808
(load all 4 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

The following 163 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St000691The number of changes of a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000971The smallest closer of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000439The position of the first down step of a 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. St001777The number of weak descents in an integer composition. St000504The cardinality of the first block of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000702The number of weak deficiencies of a permutation. St000823The number of unsplittable factors of the set partition. St000442The maximal area to the right of an up step of a Dyck path. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000874The position of the last double rise in a Dyck path. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima 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. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St001029The size of the core of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000528The height of a poset. St000527The width of the poset. St000746The number of pairs with odd minimum in a perfect matching. St000069The number of maximal elements of a poset. St000741The Colin de Verdière graph invariant. St000068The number of minimal elements in a poset. St000632The jump number of the poset. St000354The number of recoils of a permutation. St000245The number of ascents of a permutation. St000654The first descent of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000308The height of the tree associated to a permutation. St000007The number of saliances of the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000470The number of runs in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000991The number of right-to-left minima of a permutation. St000080The rank of the poset. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000989The number of final rises of a permutation. St000015The number of peaks of a Dyck path. St000087The number of induced subgraphs. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000133The "bounce" of a permutation. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000485The length of the longest cycle of a permutation. St000990The first ascent of a permutation. St001812The biclique partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001651The Frankl number of a lattice. St001948The number of augmented double ascents of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001896The number of right descents of a signed permutations. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice.