Your data matches 89 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
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Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [2,1] => [1,1,0,0]
 => 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000097
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
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
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
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: St000147
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ 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
[1,2] => ([],2)
 => [1,1]
 => 1
[2,1] => ([(0,1)],2)
 => [2]
 => 2
[1,2,3] => ([],3)
 => [1,1,1]
 => 1
[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)
 => [3]
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 2
[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)
 => [3,1]
 => 3
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 2
[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)
 => [4]
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 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)
 => [4,1]
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
Description
The largest part of an integer partition.
Matching statistic: St000172
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000381
(load all 6 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => 1
[1,2] => [1,0,1,0]
 => [1,1] => 1
[2,1] => [1,1,0,0]
 => [2] => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,2] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [3] => 3
[3,1,2] => [1,1,1,0,0,0]
 => [3] => 3
[3,2,1] => [1,1,1,0,0,0]
 => [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
Description
The largest part of an integer composition.
Matching statistic: St001029
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001108
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.
Matching statistic: St001110
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001110: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The 3-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-dynamic chromatic number of a graph.
Matching statistic: St001116
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000171The degree of the graph. St000272The treewidth of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St001120The length of a longest path in a graph. St001644The dimension of a graph. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000308The height of the tree associated to a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. 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: St001268The size of the largest ordinal summand in the poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000271The chromatic index of a graph. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000662The staircase size of the code of a permutation. St000444The length of the maximal rise of a Dyck path. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000730The maximal arc length 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. St001118The acyclic chromatic index of a graph. St001090The number of pop-stack-sorts needed to sort a permutation. St000028The number of stack-sorts needed to sort a permutation. St001652The length of a longest interval of consecutive numbers. St000651The maximal size of a rise in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001645The pebbling number of a connected graph. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000306The bounce count of a Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001717The largest size of an interval in a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000822The Hadwiger number of the graph. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001117The game chromatic index of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.