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Your data matches 273 different statistics following compositions of up to 3 maps.
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Matching statistic: St000266
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000266: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000266: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> 1
Description
The number of spanning subgraphs of a graph with the same connected components.
A subgraph or factor of a graph is spanning, if it has the same vertex set [1]. The present statistic additionally requires the subgraph to have the same components. It can be obtained by evaluating the Tutte polynomial at the points $x=1$ and $y=2$, see [2,3].
By mistake, [2] refers to this statistic as the number of spanning subgraphs, which would be $2^m$, where $m$ is the number of edges. Equivalently, this would be the evaluation of the Tutte polynomial at $x=y=2$.
Matching statistic: St000637
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000637: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000637: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The length of the longest cycle in a graph.
This statistic is zero for acyclic graphs.
Matching statistic: St001478
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001478: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001478: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 0 = 1 - 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The number of nowhere zero 4-flows of a graph.
Matching statistic: St000270
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 1 + 3
[3,1,2] => ([(0,2),(1,2)],3)
=> 4 = 1 + 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 4 = 1 + 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 4 = 1 + 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 1 + 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 4 = 1 + 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 4 = 1 + 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 7 = 4 + 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4 = 1 + 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 7 = 4 + 3
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4 = 1 + 3
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4 = 1 + 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 4 = 1 + 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 7 = 4 + 3
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 7 = 4 + 3
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 7 = 4 + 3
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 7 = 4 + 3
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 4 = 1 + 3
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 4 = 1 + 3
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 7 = 4 + 3
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 4 = 1 + 3
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> 4 = 1 + 3
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 7 = 4 + 3
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> 4 = 1 + 3
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> 4 = 1 + 3
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> 4 = 1 + 3
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> 7 = 4 + 3
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> 4 = 1 + 3
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> 4 = 1 + 3
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> 4 = 1 + 3
Description
The number of forests contained in a graph.
That is, for a graph $G = (V,E)$ with vertices $V$ and edges $E$, the number of subsets $E' \subseteq E$ for which the subgraph $(V,E')$ is acyclic.
If $T_G(x,y)$ is the Tutte polynomial [2] of $G$, then the number of forests contained in $G$ is given by $T_G(2,1)$.
Matching statistic: St000207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> [3]
=> 4
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> [3]
=> 4
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000468
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St000714
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> [1,1]
=> 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> [1,1]
=> 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> [3]
=> 4
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> [3]
=> 4
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> [1,1]
=> 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> [1,1]
=> 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St001477
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],4)
=> 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ([],4)
=> 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 4
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([],5)
=> 1
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([],5)
=> 1
Description
The number of nowhere zero 5-flows of a graph.
Matching statistic: St001562
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001562: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001562: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [1,1,1]
=> [1,1]
=> 4
[1,3,4,2] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 4
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 4
[1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 4
[1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 4
[2,1,3,5,4] => [3,2]
=> [2]
=> 1
[2,1,4,3,5] => [3,2]
=> [2]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> 1
[3,1,2,4,5] => [4,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 4
[1,2,3,5,6,4] => [5,1]
=> [1]
=> 1
[1,2,3,6,4,5] => [5,1]
=> [1]
=> 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 4
[1,2,4,3,6,5] => [4,2]
=> [2]
=> 1
[1,2,4,5,3,6] => [5,1]
=> [1]
=> 1
[1,2,5,3,4,6] => [5,1]
=> [1]
=> 1
[1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 4
[1,3,2,4,6,5] => [4,2]
=> [2]
=> 1
[1,3,2,5,4,6] => [4,2]
=> [2]
=> 1
[1,3,4,2,5,6] => [5,1]
=> [1]
=> 1
[1,4,2,3,5,6] => [5,1]
=> [1]
=> 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,1]
=> 4
[2,1,3,4,6,5] => [4,2]
=> [2]
=> 1
[2,1,3,5,4,6] => [4,2]
=> [2]
=> 1
[2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[2,3,1,4,5,6] => [5,1]
=> [1]
=> 1
[3,1,2,4,5,6] => [5,1]
=> [1]
=> 1
[3,2,1,4,5,6] => [4,1,1]
=> [1,1]
=> 4
[1,2,3,4,6,7,5] => [6,1]
=> [1]
=> 1
[1,2,3,4,7,5,6] => [6,1]
=> [1]
=> 1
[1,2,3,4,7,6,5] => [5,1,1]
=> [1,1]
=> 4
[1,2,3,5,4,7,6] => [5,2]
=> [2]
=> 1
[1,2,3,5,6,4,7] => [6,1]
=> [1]
=> 1
[1,2,3,6,4,5,7] => [6,1]
=> [1]
=> 1
[1,2,3,6,5,4,7] => [5,1,1]
=> [1,1]
=> 4
[1,2,4,3,5,7,6] => [5,2]
=> [2]
=> 1
[1,2,4,3,6,5,7] => [5,2]
=> [2]
=> 1
[1,2,4,5,3,6,7] => [6,1]
=> [1]
=> 1
Description
The value of the complete homogeneous symmetric function evaluated at 1.
The statistic is $h_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$,
where $\lambda$ has $k$ parts.
Matching statistic: St001563
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001563: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001563: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [1,1,1]
=> [1,1]
=> 4
[1,3,4,2] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 4
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 4
[1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 4
[1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 4
[2,1,3,5,4] => [3,2]
=> [2]
=> 1
[2,1,4,3,5] => [3,2]
=> [2]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> 1
[3,1,2,4,5] => [4,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 4
[1,2,3,5,6,4] => [5,1]
=> [1]
=> 1
[1,2,3,6,4,5] => [5,1]
=> [1]
=> 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 4
[1,2,4,3,6,5] => [4,2]
=> [2]
=> 1
[1,2,4,5,3,6] => [5,1]
=> [1]
=> 1
[1,2,5,3,4,6] => [5,1]
=> [1]
=> 1
[1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 4
[1,3,2,4,6,5] => [4,2]
=> [2]
=> 1
[1,3,2,5,4,6] => [4,2]
=> [2]
=> 1
[1,3,4,2,5,6] => [5,1]
=> [1]
=> 1
[1,4,2,3,5,6] => [5,1]
=> [1]
=> 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,1]
=> 4
[2,1,3,4,6,5] => [4,2]
=> [2]
=> 1
[2,1,3,5,4,6] => [4,2]
=> [2]
=> 1
[2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[2,3,1,4,5,6] => [5,1]
=> [1]
=> 1
[3,1,2,4,5,6] => [5,1]
=> [1]
=> 1
[3,2,1,4,5,6] => [4,1,1]
=> [1,1]
=> 4
[1,2,3,4,6,7,5] => [6,1]
=> [1]
=> 1
[1,2,3,4,7,5,6] => [6,1]
=> [1]
=> 1
[1,2,3,4,7,6,5] => [5,1,1]
=> [1,1]
=> 4
[1,2,3,5,4,7,6] => [5,2]
=> [2]
=> 1
[1,2,3,5,6,4,7] => [6,1]
=> [1]
=> 1
[1,2,3,6,4,5,7] => [6,1]
=> [1]
=> 1
[1,2,3,6,5,4,7] => [5,1,1]
=> [1,1]
=> 4
[1,2,4,3,5,7,6] => [5,2]
=> [2]
=> 1
[1,2,4,3,6,5,7] => [5,2]
=> [2]
=> 1
[1,2,4,5,3,6,7] => [6,1]
=> [1]
=> 1
Description
The value of the power-sum symmetric function evaluated at 1.
The statistic is $p_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$,
where $\lambda$ has $k$ parts.
The following 263 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001877Number of indecomposable injective modules with projective dimension 2. St000081The number of edges of a graph. St000088The row sums of the character table of the symmetric group. St000226The convexity of a permutation. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000313The number of degree 2 vertices of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001077The prefix exchange distance of a permutation. St001279The sum of the parts of an integer partition that are at least two. St001341The number of edges in the center of a graph. St001375The pancake length of a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001541The Gini index of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000145The Dyson rank of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000345The number of refinements of a partition. St000869The sum of the hook lengths of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000146The Andrews-Garvan crank of a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001833The number of linear intervals in a lattice. St000712The number of semistandard Young tableau of given shape, with entries at most 4. 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. St001129The product of the squares of the parts of a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000148The number of odd parts of a partition. St000185The weighted size of a partition. St000475The number of parts equal to 1 in a partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000753The Grundy value for the game of Kayles on a binary word. St000937The number of positive values of the symmetric group character corresponding to the partition. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001961The sum of the greatest common divisors of all pairs of parts. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000762The sum of the positions of the weak records of an integer composition. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000086The number of subgraphs. St001117The game chromatic index of a graph. St001649The length of a longest trail in a graph. St001850The number of Hecke atoms of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001651The Frankl number of a lattice. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000284The Plancherel distribution on integer partitions. St000327The number of cover relations in a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000634The number of endomorphisms of a poset. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000422The energy of a graph, if it is integral. St000477The weight of a partition according to Alladi. St000302The determinant of the distance matrix of a connected graph. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000959The number of strong Bruhat factorizations of a permutation. St000309The number of vertices with even degree. St000972The composition number of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St000351The determinant of the adjacency matrix of a graph. St001531Number of partial orders contained in the poset determined by the Dyck path. St000095The number of triangles of a graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St000274The number of perfect matchings of a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001312Number of parabolic noncrossing partitions indexed by the composition. St001395The number of strictly unfriendly partitions of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000456The monochromatic index of a connected graph. St000867The sum of the hook lengths in the first row of an integer partition. St001694The number of maximal dissociation sets in a graph. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000741The Colin de Verdière graph invariant. St000978The sum of the positions of double down-steps of a Dyck path. St001330The hat guessing number of a graph. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St000781The number of proper colouring schemes of a Ferrers diagram. St000908The length of the shortest maximal antichain in a poset. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. 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. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001964The interval resolution global dimension of a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000189The number of elements in the poset. St001717The largest size of an interval in a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St000699The toughness times the least common multiple of 1,. 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. St000464The Schultz index of a connected graph. 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 $eA$. 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 $eA$. 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 $eA$. St001545The second Elser number of a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001851The number of Hecke atoms of a signed permutation. St000708The product of the parts of an integer partition. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer 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. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees 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. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000474Dyson's crank of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.
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