Your data matches 326 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000062
(load all 24 compositions to match this statistic)
Mapping
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 3
[1,4,5,2,3] => 3
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000451
Mapping
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 3
[3,4,2,1] => 2
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 3
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 3
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000010
(load all 6 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ 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] => [2]
 => 1
[2,1] => [1,1]
 => 2
[1,2,3] => [3]
 => 1
[1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1]
 => 2
[2,3,1] => [2,1]
 => 2
[3,1,2] => [2,1]
 => 2
[3,2,1] => [1,1,1]
 => 3
[1,2,3,4] => [4]
 => 1
[1,2,4,3] => [3,1]
 => 2
[1,3,2,4] => [3,1]
 => 2
[1,3,4,2] => [3,1]
 => 2
[1,4,2,3] => [3,1]
 => 2
[1,4,3,2] => [2,1,1]
 => 3
[2,1,3,4] => [3,1]
 => 2
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => 2
[2,3,4,1] => [3,1]
 => 2
[2,4,1,3] => [2,2]
 => 2
[2,4,3,1] => [2,1,1]
 => 3
[3,1,2,4] => [3,1]
 => 2
[3,1,4,2] => [2,2]
 => 2
[3,2,1,4] => [2,1,1]
 => 3
[3,2,4,1] => [2,1,1]
 => 3
[3,4,1,2] => [2,2]
 => 2
[3,4,2,1] => [2,1,1]
 => 3
[4,1,2,3] => [3,1]
 => 2
[4,1,3,2] => [2,1,1]
 => 3
[4,2,1,3] => [2,1,1]
 => 3
[4,2,3,1] => [2,1,1]
 => 3
[4,3,1,2] => [2,1,1]
 => 3
[4,3,2,1] => [1,1,1,1]
 => 4
[1,2,3,4,5] => [5]
 => 1
[1,2,3,5,4] => [4,1]
 => 2
[1,2,4,3,5] => [4,1]
 => 2
[1,2,4,5,3] => [4,1]
 => 2
[1,2,5,3,4] => [4,1]
 => 2
[1,2,5,4,3] => [3,1,1]
 => 3
[1,3,2,4,5] => [4,1]
 => 2
[1,3,2,5,4] => [3,2]
 => 2
[1,3,4,2,5] => [4,1]
 => 2
[1,3,4,5,2] => [4,1]
 => 2
[1,3,5,2,4] => [3,2]
 => 2
[1,3,5,4,2] => [3,1,1]
 => 3
[1,4,2,3,5] => [4,1]
 => 2
[1,4,2,5,3] => [3,2]
 => 2
[1,4,3,2,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => 3
[1,4,5,2,3] => [3,2]
 => 2
Description
The length of the partition.
Matching statistic: St000093
(load all 9 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 2
[2,1] => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => 3
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 4
[1,2,4,3] => ([(2,3)],4)
 => 3
[1,3,2,4] => ([(2,3)],4)
 => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4,5] => ([],5)
 => 5
[1,2,3,5,4] => ([(3,4)],5)
 => 4
[1,2,4,3,5] => ([(3,4)],5)
 => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000097
(load all 9 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 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 9 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 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: St000147
(load all 9 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 1
[1,2] => [2]
 => 2
[2,1] => [1,1]
 => 1
[1,2,3] => [3]
 => 3
[1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1]
 => 2
[2,3,1] => [2,1]
 => 2
[3,1,2] => [2,1]
 => 2
[3,2,1] => [1,1,1]
 => 1
[1,2,3,4] => [4]
 => 4
[1,2,4,3] => [3,1]
 => 3
[1,3,2,4] => [3,1]
 => 3
[1,3,4,2] => [3,1]
 => 3
[1,4,2,3] => [3,1]
 => 3
[1,4,3,2] => [2,1,1]
 => 2
[2,1,3,4] => [3,1]
 => 3
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => 3
[2,3,4,1] => [3,1]
 => 3
[2,4,1,3] => [2,2]
 => 2
[2,4,3,1] => [2,1,1]
 => 2
[3,1,2,4] => [3,1]
 => 3
[3,1,4,2] => [2,2]
 => 2
[3,2,1,4] => [2,1,1]
 => 2
[3,2,4,1] => [2,1,1]
 => 2
[3,4,1,2] => [2,2]
 => 2
[3,4,2,1] => [2,1,1]
 => 2
[4,1,2,3] => [3,1]
 => 3
[4,1,3,2] => [2,1,1]
 => 2
[4,2,1,3] => [2,1,1]
 => 2
[4,2,3,1] => [2,1,1]
 => 2
[4,3,1,2] => [2,1,1]
 => 2
[4,3,2,1] => [1,1,1,1]
 => 1
[1,2,3,4,5] => [5]
 => 5
[1,2,3,5,4] => [4,1]
 => 4
[1,2,4,3,5] => [4,1]
 => 4
[1,2,4,5,3] => [4,1]
 => 4
[1,2,5,3,4] => [4,1]
 => 4
[1,2,5,4,3] => [3,1,1]
 => 3
[1,3,2,4,5] => [4,1]
 => 4
[1,3,2,5,4] => [3,2]
 => 3
[1,3,4,2,5] => [4,1]
 => 4
[1,3,4,5,2] => [4,1]
 => 4
[1,3,5,2,4] => [3,2]
 => 3
[1,3,5,4,2] => [3,1,1]
 => 3
[1,4,2,3,5] => [4,1]
 => 4
[1,4,2,5,3] => [3,2]
 => 3
[1,4,3,2,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => 3
[1,4,5,2,3] => [3,2]
 => 3
Description
The largest part of an integer partition.
Matching statistic: St000308
(load all 13 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [1,3,2] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 3
[1,3,4,2] => [1,2,4,3] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 3
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => 3
[2,3,4,1] => [1,2,4,3] => 3
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => 3
[3,1,4,2] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 4
[1,2,4,5,3] => [1,2,3,5,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => 4
[1,3,2,5,4] => [1,3,2,5,4] => 3
[1,3,4,2,5] => [1,2,4,3,5] => 4
[1,3,4,5,2] => [1,2,3,5,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => 3
[1,3,5,4,2] => [1,2,5,4,3] => 3
[1,4,2,3,5] => [1,4,2,3,5] => 4
[1,4,2,5,3] => [1,3,2,5,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => [1,3,2,5,4] => 3
[1,4,5,2,3] => [1,3,5,2,4] => 3
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000527
(load all 4 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000527: Posets ⟶ 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
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 3
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 3
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(2,3)],4)
 => 3
[4,3,1,2] => ([(2,3)],4)
 => 3
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000528
(load all 6 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => 2
[4,3,2,1] => ([],4)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
The following 316 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000080The rank of the poset. St000007The number of saliances of the permutation. St000087The number of induced subgraphs. St000105The number of blocks in the set partition. 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. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. 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. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 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. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000245The number of ascents of a permutation. 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. 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. 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. 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. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000141The maximum drop size of a permutation. St000184The size of the centralizer of any permutation of given cycle type. St000213The number of weak exceedances (also weak excedences) of a permutation. St000228The size of a partition. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000383The last part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000443The number of long tunnels of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000505The biggest entry in the block containing the 1. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000667The greatest common divisor of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. 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. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000971The smallest closer of a set partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant 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. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001530The depth of a Dyck path. St001571The Cartan determinant of the integer partition. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000008The major index of the composition. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000133The "bounce" of a permutation. St000145The Dyson rank of a partition. St000155The number of exceedances (also excedences) of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000446The disorder of a permutation. St000532The total number of rook placements on a Ferrers board. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. 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. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000806The semiperimeter of the associated bargraph. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000061The number of nodes on the left branch of a binary tree. 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. St000485The length of the longest cycle of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001834The number of non-isomorphic minors of a graph. St001706The number of closed sets in 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. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000006The dinv of a Dyck path. 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. St001323The independence gap of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000710The number of big deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000291The number of descents of a binary word. St000143The largest repeated part of a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001948The number of augmented double ascents of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001863The number of weak excedances of a signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$.