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Matching statistic: St000010
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Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
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%
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
[2,3,1] => [3,1,2] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [1,1,1]
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => [2,1]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [3,1]
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => [3,1]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [3,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => [3,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [4,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [4,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [4,1]
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => [4,1]
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => [4,1]
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [5,1]
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [5,1]
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [5,1]
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [5,1]
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [5,1]
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [4,1,1]
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => [5,1]
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2]
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [4,2]
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [4,2]
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => [5,1]
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => [4,1,1]
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => [5,1]
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [6,1]
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [5,1,1]
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,1]
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,2]
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [6,1]
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [5,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [6,1]
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [5,2]
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [5,2]
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [6,1]
=> 2 = 3 - 1
Description
The length of the partition.
Matching statistic: St000097
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [3]
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => [2,1]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [3,1]
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => [3,1]
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [3,1,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [3,1,1,1]
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,1,1]
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [3,1,1,1]
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,2,1,1]
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,2,1,1]
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => [3,1,1,1]
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [3,1,1,1,1]
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [3,1,1,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
Description
The largest part of an integer partition.
Matching statistic: St000172
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
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: St000453
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,4,2] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,3,5,4,6,7] => ([(5,6)],7)
=> 2 = 3 - 1
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000668
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [3]
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => [2,1]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [3,1]
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => [3,1]
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [3,1,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [3,1,1,1]
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,1,1]
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [3,1,1,1]
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,2,1,1]
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,2,1,1]
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,2,1,1]
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => [3,1,1,1]
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => [2,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [3,1,1,1,1]
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [3,1,1,1,1]
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,2,1,1,1]
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [2,1,1,1,1,1]
=> 2 = 3 - 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000972
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000972: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000972: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 5 = 4 + 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 3 + 1
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 4 = 3 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 4 = 3 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 4 = 3 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The composition number of a graph.
This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.
Matching statistic: St001029
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 2 = 3 - 1
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: St001062
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2 = 3 - 1
[3,1,2] => [2,3,1] => {{1,2,3}}
=> 3 = 4 - 1
[3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 3 = 4 - 1
[1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 2 = 3 - 1
[3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 3 = 4 - 1
[3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[1,4,2,3,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3 = 4 - 1
[1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 3 - 1
[2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 3 - 1
[3,1,2,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3 = 4 - 1
[3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 3 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => {{1},{2},{3},{4,6},{5}}
=> 2 = 3 - 1
[1,2,3,6,4,5] => [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}}
=> 3 = 4 - 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}}
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => {{1},{2},{3,4},{5,6}}
=> 2 = 3 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => {{1},{2},{3,5},{4},{6}}
=> 2 = 3 - 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => {{1},{2},{3,4,5},{6}}
=> 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}}
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => {{1},{2,3},{4},{5,6}}
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => {{1},{2,3},{4,5},{6}}
=> 2 = 3 - 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => {{1},{2,4},{3},{5},{6}}
=> 2 = 3 - 1
[1,4,2,3,5,6] => [1,3,4,2,5,6] => {{1},{2,3,4},{5},{6}}
=> 3 = 4 - 1
[1,4,3,2,5,6] => [1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}}
=> 2 = 3 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => {{1,2},{3},{4},{5,6}}
=> 2 = 3 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => {{1,2},{3},{4,5},{6}}
=> 2 = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}}
=> 2 = 3 - 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => {{1,3},{2},{4},{5},{6}}
=> 2 = 3 - 1
[3,1,2,4,5,6] => [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}}
=> 3 = 4 - 1
[3,2,1,4,5,6] => [3,2,1,4,5,6] => {{1,3},{2},{4},{5},{6}}
=> 2 = 3 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,7},{6}}
=> 2 = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> 3 = 4 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> 2 = 3 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}}
=> 2 = 3 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => {{1},{2},{3},{4,6},{5},{7}}
=> 2 = 3 - 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}}
=> 3 = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> 2 = 3 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}}
=> 2 = 3 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}}
=> 2 = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => {{1},{2},{3,5},{4},{6},{7}}
=> 2 = 3 - 1
Description
The maximal size of a block of a set partition.
The following 825 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001093The detour number of a graph. St001116The game chromatic number of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000201The number of leaf nodes in a binary tree. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000679The pruning number of an ordered tree. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001568The smallest positive integer that does not appear twice in the partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001732The number of peaks visible from the left. St001743The discrepancy of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001962The proper pathwidth of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000344The number of strongly connected outdegree sequences of a graph. St000347The inversion sum of a binary word. St000386The number of factors DDU in a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000552The number of cut vertices of a graph. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001172The number of 1-rises at odd height of a Dyck path. St001308The number of induced paths on three 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. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001350Half of the Albertson index of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001736The total number of cycles in a graph. St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000459The hook length of the base cell of a partition. St000521The number of distinct subtrees of an ordered tree. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St001809The index of the step at the first peak of maximal height in a Dyck path. St001875The number of simple modules with projective dimension at most 1. St000008The major index of the composition. St000012The area of a Dyck path. St000013The height of a Dyck path. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000288The number of ones in a binary word. St000330The (standard) major index of a standard tableau. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000391The sum of the positions of the ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000636The hull number of a graph. 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. St000708The product of the parts of an integer partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000792The Grundy value for the game of ruler on a binary word. St000822The Hadwiger number of the graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000874The position of the last double rise in a Dyck path. St000926The clique-coclique number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St000984The number of boxes below precisely one peak. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001706The number of closed sets in a graph. St001717The largest size of an interval in a poset. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000040The number of regions of the inversion arrangement of a permutation. St000048The multinomial of the parts of a partition. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000377The dinv defect of an integer partition. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000454The largest eigenvalue of a graph if it is integral. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000526The number of posets with combinatorially isomorphic order polytopes. St000527The width of the poset. St000548The number of different non-empty partial sums of an integer partition. St000568The hook number of a binary tree. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000741The Colin de Verdière graph invariant. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000753The Grundy value for the game of Kayles on a binary word. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000778The metric dimension of a graph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000844The size of the largest block in the direct sum decomposition of a permutation. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000913The number of ways to refine the partition into singletons. St000920The logarithmic height of a Dyck path. St000988The orbit size of a permutation under Foata's bijection. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001120The length of a longest path in a graph. St001246The maximal difference between two consecutive entries of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001345The Hamming dimension 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001619The number of non-isomorphic sublattices of a lattice. St001625The Möbius invariant of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001716The 1-improper chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001777The number of weak descents in an integer composition. St001933The largest multiplicity of a part in an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001949The rigidity index of a graph. St000057The Shynar inversion number of a standard tableau. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000137The Grundy value of an integer partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000209Maximum difference of elements in cycles. St000223The number of nestings in the permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000369The dinv deficit of a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000387The matching number of a graph. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000463The number of admissible inversions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000539The number of odd inversions of a permutation. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000624The normalized sum of the minimal distances to a greater element. St000632The jump number of the poset. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000884The number of isolated descents of a permutation. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000974The length of the trunk of an ordered tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001280The number of parts of an integer partition that are at least two. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. 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. St001354The number of series nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001383The BG-rank of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001394The genus of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001618The cardinality of the Frattini sublattice of a lattice. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001718The number of non-empty open intervals in a poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001961The sum of the greatest common divisors of all pairs of parts. St000997The even-odd crank of an integer partition. St001826The maximal number of leaves on a vertex of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000299The number of nonisomorphic vertex-induced subtrees. St000546The number of global descents of a permutation. St000961The shifted major index of a permutation. St000989The number of final rises of a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000360The number of occurrences of the pattern 32-1. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St000234The number of global ascents of a permutation. St000833The comajor index of a permutation. St000355The number of occurrences of the pattern 21-3. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000367The number of simsun double descents of a permutation. St001850The number of Hecke atoms of a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000081The number of edges of a graph. St000264The girth of a graph, which is not a tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St001115The number of even descents of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000002The number of occurrences of the pattern 123 in a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000646The number of big ascents of a permutation. St000665The number of rafts of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001586The number of odd parts smaller than the largest even part in an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001396Number of triples of incomparable elements in a finite poset. St000849The number of 1/3-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000441The number of successions of a permutation. St000308The height of the tree associated to a permutation. St001377The major index minus the number of inversions of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001645The pebbling number of a connected graph. St000255The number of reduced Kogan faces with the permutation as type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St000990The first ascent of a permutation. St000245The number of ascents of a permutation. St001427The number of descents of a signed permutation. St001114The number of odd descents of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000246The number of non-inversions of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001077The prefix exchange distance of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000462The major index minus the number of excedences of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000224The sorting index of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000086The number of subgraphs. St000004The major index of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000067The inversion number of the alternating sign matrix. St000325The width of the tree associated to a permutation. St000332The positive inversions of an alternating sign matrix. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001428The number of B-inversions of a signed permutation. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000889The number of alternating sign matrices with the same antidiagonal sums. St000023The number of inner peaks of a permutation. St000357The number of occurrences of the pattern 12-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000756The sum of the positions of the left to right maxima of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000166The depth minus 1 of an ordered tree. St000238The number of indices that are not small weak excedances. St000286The number of connected components of the complement of a graph. St000305The inverse major index of a permutation. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000397The Strahler number of a rooted tree. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000722The number of different neighbourhoods in a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. St001341The number of edges in the center of a graph. St001346The number of parking functions that give the same permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001530The depth of a Dyck path. 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. St000045The number of linear extensions of a binary tree. St000056The decomposition (or block) number of a permutation. St000080The rank of the poset. St000109The number of elements less than or equal to the given element in Bruhat order. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000362The size of a minimal vertex cover of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000472The sum of the ascent bottoms of a permutation. St000482The (zero)-forcing number of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000659The number of rises of length at least 2 of a Dyck path. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000948The chromatic discriminant of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001119The length of a shortest maximal path in a graph. 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 St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000216The absolute length of a permutation. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000523The number of 2-protected nodes of a rooted tree. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001546The number of monomials in the Tutte polynomial of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001948The number of augmented double ascents of a permutation. St000422The energy of a graph, if it is integral. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000707The product of the factorials of the parts. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001877Number of indecomposable injective modules with projective dimension 2. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001060The distinguishing index of a graph. St000806The semiperimeter of the associated bargraph. St001118The acyclic chromatic index of a graph. St001378The product of the cohook lengths of the integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001943The sum of the squares of the hook lengths of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001525The number of symmetric hooks on the diagonal of a partition. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001527The cyclic permutation representation number of an integer partition. St001898The number of occurrences of an 132 pattern in an ordered set partition. 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 $eA$. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001811The Castelnuovo-Mumford regularity of a permutation. St001555The order of a signed permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001684The reduced word complexity of a permutation. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000699The toughness times the least common multiple of 1,. St000464The Schultz index of a connected 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 $eA$. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St001199The dominant dimension of $eAe$ for 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001864The number of excedances of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000145The Dyson rank of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St000474Dyson's crank of a partition. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001867The number of alignments of type EN of a signed permutation. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001868The number of alignments of type NE of a signed permutation. St001821The sorting index of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001895The oddness of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001624The breadth of a lattice.
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