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Your data matches 204 different statistics following compositions of up to 3 maps.
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Matching statistic: St000822
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Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> 2
([(0,1)],2)
=> ([],2)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
Description
The Hadwiger number of the graph.
Also known as clique contraction number, this is the size of the largest complete minor.
Matching statistic: St001580
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Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> 2
([(0,1)],2)
=> ([],2)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
Description
The acyclic chromatic number of a graph.
This is the smallest size of a vertex partition $\{V_1,\dots,V_k\}$ such that each $V_i$ is an independent set and for all $i,j$ the subgraph inducted by $V_i\cup V_j$ does not contain a cycle.
Matching statistic: St000272
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Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000536
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Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The pathwidth of a graph.
Matching statistic: St001277
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(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
Matching statistic: St001358
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(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The largest degree of a regular subgraph of a graph.
For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].
Matching statistic: St001792
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The arboricity of a graph.
This is the minimum number of forests that covers all edges of the graph.
Matching statistic: St000378
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000378: 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
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> 0 = 1 - 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St001092
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],2)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [1]
=> [1]
=> 0 = 1 - 1
([(1,2)],3)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [2]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4]
=> [2,2]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [4,2]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [4,1]
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [2,2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [2,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [4,2]
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [2,2]
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [2]
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [5,5]
=> [4,3,2,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [4,1]
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [2,2,1]
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> [1]
=> 0 = 1 - 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001331
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1)],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
The following 194 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001484The number of singletons of an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001345The Hamming dimension of a graph. St001330The hat guessing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St000015The number of peaks of a Dyck path. St000120The number of left tunnels of a Dyck path. St000159The number of distinct parts of the integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001432The order dimension of the partition. St001530The depth of a Dyck path. St000481The number of upper covers of a partition in dominance order. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000527The width of the poset. St001720The minimal length of a chain of small intervals in a lattice. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001280The number of parts of an integer partition that are at least two. St001739The number of graphs with the same edge polytope as the given graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000671The maximin edge-connectivity for choosing a subgraph. St001139The number of occurrences of hills of size 2 in a Dyck path. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the 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 $eA$. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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 $eA$. St001651The Frankl number of a lattice. St000260The radius of a connected graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001624The breadth of a lattice. St000544The cop number of a graph. St001029The size of the core of a graph. St001270The bandwidth of a graph. St001494The Alon-Tarsi number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000537The cutwidth of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. 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. St000455The second largest eigenvalue of a graph if it is integral. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000454The largest eigenvalue of a graph if it is integral. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000640The rank of the largest boolean interval in a poset. St000264The girth of a graph, which is not a tree. St000307The number of rowmotion orbits of a poset. St000741The Colin de Verdière graph invariant. St001734The lettericity of a 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. St001642The Prague dimension of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001118The acyclic chromatic index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. 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. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. 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. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer 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. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001610The number of coloured endofunctions 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. St001627The number of coloured connected graphs 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. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. 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. 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. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000699The toughness times the least common multiple of 1,. 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the 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. St001541The Gini index of an 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. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition.
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