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Your data matches 282 different statistics following compositions of up to 3 maps.
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Matching statistic: St000266
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> 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: St000313
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
Description
The number of degree 2 vertices of a graph.
A vertex has degree 2 if and only if it lies on a unique maximal path.
Matching statistic: St000637
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> 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 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
Description
The number of nowhere zero 4-flows of a graph.
Matching statistic: St000270
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> 4 = 1 + 3
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 7 = 4 + 3
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> 4 = 1 + 3
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> 7 = 4 + 3
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> 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: St000086
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
Description
The number of subgraphs.
Given a graph $G$, this is the number of graphs $H$ such that $H \hookrightarrow G$.
Matching statistic: St000207
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> [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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St000714
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> [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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
Description
The number of nowhere zero 5-flows of a graph.
The following 272 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. St000096The number of spanning trees of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index 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. St001117The game chromatic index of a graph. St001279The sum of the parts of an integer partition that are at least two. St001341The number of edges in the center of a graph. 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. St001649The length of a longest trail in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000145The Dyson rank of a partition. St000869The sum of the hook lengths of an 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. St000309The number of vertices with even degree. St000972The composition number of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St000302The determinant of the distance matrix of a connected graph. St000351The determinant of the adjacency matrix of a graph. St000762The sum of the positions of the weak records of an integer composition. St001563The value of the power-sum symmetric function evaluated at 1. St001249Sum of the odd parts of a partition. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001645The pebbling number of a connected graph. St000467The hyper-Wiener index of a connected graph. 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. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St000455The second largest eigenvalue of a graph if it is integral. St000464The Schultz index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001618The cardinality of the Frattini sublattice of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001395The number of strictly unfriendly partitions of a graph. St000079The number of alternating sign matrices for a given 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. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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. St001592The maximal number of simple paths between any two different vertices of a graph. St001623The number of doubly irreducible elements of a lattice. St001706The number of closed sets in a graph. St001827The number of two-component spanning forests of a graph. St000479The Ramsey number of a graph. St000867The sum of the hook lengths in the first row of an integer partition. St001619The number of non-isomorphic sublattices of a lattice. St001625The Möbius invariant of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001694The number of maximal dissociation sets in a graph. St001931The weak major index of an integer composition regarded as a word. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000284The Plancherel distribution on integer partitions. St000361The second Zagreb index of a graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000770The major index of an integer partition when read from bottom to top. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. 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. St001129The product of the squares of the parts of a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001764The number of non-convex subsets of vertices in a graph. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000185The weighted size of a partition. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000946The sum of the skew hook positions in a Dyck path. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000978The sum of the positions of double down-steps of a Dyck path. 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001127The sum of the squares of the parts of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001360The number of covering relations in Young's lattice below a partition. St001391The disjunction number of a graph. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St000063The number of linear extensions of a certain poset defined for an integer partition. St000294The number of distinct factors of a binary word. St000347The inversion sum of a binary word. St000391The sum of the positions of the ones in a binary word. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001441The number of non-empty connected induced subgraphs of a graph. St001564The value of the forgotten symmetric functions when all variables set to 1. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001930The weak major index of a binary word. St000997The even-odd crank of an integer partition. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001762The number of convex subsets of vertices in a graph. St001834The number of non-isomorphic minors of a graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St000699The toughness times the least common multiple of 1,. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000379The number of Hamiltonian cycles in a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 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. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. 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. 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. 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. 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. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. 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. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer 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 $eA$. 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 $eA$. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. 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. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001281The normalized isoperimetric number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000567The sum of the products of all pairs of parts. St000940The number of characters of the symmetric group whose value on the partition is zero. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001330The hat guessing number of a graph. St000639The number of relations in a poset. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000068The number of minimal elements in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001301The first Betti number of the order complex associated with the poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.
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