- FindStat

Your data matches 261 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000528

Mapping

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

Values

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

Description

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

Matching statistic: St000080

Mapping

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

Values

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

Description

The rank of the poset.

Matching statistic: St000093
(load all 3 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ 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
([],2)
 => [1,1]
 => [2]
 => 1
([(0,1)],2)
 => [2]
 => [1,1]
 => 2
([],3)
 => [1,1,1]
 => [3]
 => 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => 2

Description

The length of the partition.

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

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$G$ is a subset

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

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

$U$ is a complete graph.

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

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

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

Mapping

Mp00198: Posets —incomparability graph⟶ 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
([],2)
 => ([(0,1)],2)
 => [1,1] => 1
([(0,1)],2)
 => ([],2)
 => [2] => 2
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,1] => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1] => 2
([(0,2),(2,1)],3)
 => ([],3)
 => [3] => 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1] => 2
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1] => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2] => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [4] => 4
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1] => 3
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [4,1] => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [3,2] => 3
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [3,2] => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 2

Description

The largest part of an integer composition.

Matching statistic: St000382

Mapping

Mp00198: Posets —incomparability graph⟶ 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
([],2)
 => ([(0,1)],2)
 => [1,1] => 1
([(0,1)],2)
 => ([],2)
 => [2] => 2
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,1] => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1] => 2
([(0,2),(2,1)],3)
 => ([],3)
 => [3] => 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1] => 2
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1] => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [3,1] => 3
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2] => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [4] => 4
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1] => 3
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [4,1] => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [3,2] => 3
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [3,2] => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 2

Description

The first part of an integer composition.

Matching statistic: St000676
(load all 2 compositions to match this statistic)

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ 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
([],2)
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2

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].

The following 251 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St001029The size of the core of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000141The maximum drop size of a permutation. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000636The hull number of a graph. St000691The number of changes of a binary word. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. 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. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000439The position of the first down step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000741The Colin de Verdière graph invariant. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001777The number of weak descents in an integer composition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St000504The cardinality of the first block of a set partition. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001062The maximal size of a block of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000874The position of the last double rise in a Dyck path. St000989The number of final rises of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number 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. 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. St000746The number of pairs with odd minimum in a perfect matching. St001674The number of vertices of the largest induced star graph in the 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000143The largest repeated part of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

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

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001471The magnitude of a Dyck path. St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000306The bounce count of a Dyck path. St000632The jump number of the poset. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. 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). St000015The number of peaks of a Dyck path. St000120The number of left tunnels of a Dyck path. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000144The pyramid weight of the Dyck path. St000331The number of upper interactions of a Dyck path. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001432The order dimension of the partition. 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

$eA$ . St000783The side length of the largest staircase partition fitting into a partition. St001488The number of corners of a skew partition. St001720The minimal length of a chain of small intervals in a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000387The matching number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000537The cutwidth of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000785The number of distinct colouring schemes of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001792The arboricity of a graph. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001393The induced matching number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph.