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Your data matches 377 different statistics following compositions of up to 3 maps.
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Matching statistic: St000093
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([],3)
=> 3 = -1 + 4
([],4)
=> 4 = 0 + 4
([(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St001337
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([],3)
=> 3 = -1 + 4
([],4)
=> 4 = 0 + 4
([(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([],3)
=> 3 = -1 + 4
([],4)
=> 4 = 0 + 4
([(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St001323
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = -1 + 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = -1 + 3
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = -1 + 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = -1 + 3
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(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 = -1 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = -1 + 3
([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(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)
=> 3 = 0 + 3
([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(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)
=> 3 = 0 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 3 = 0 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
Description
The independence gap of a graph.
This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph.
In particular, this statistic is $0$ for well covered graphs
Matching statistic: St000097
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = -1 + 4
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 0 + 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = -1 + 4
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 0 + 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000381
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [3] => 3 = -1 + 4
([],4)
=> [4] => 4 = 0 + 4
([(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(1,3),(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(1,4),(2,3)],5)
=> [3,2] => 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(2,5),(3,4)],6)
=> [4,2] => 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [3] => 3 = -1 + 4
([],4)
=> [4] => 4 = 0 + 4
([(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(1,3),(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3 = -1 + 4
([(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 0 + 4
([(1,4),(2,3)],5)
=> [3,2] => 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3 = -1 + 4
([(2,5),(3,4)],6)
=> [4,2] => 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 4 = 0 + 4
Description
The first part of an integer composition.
Matching statistic: St001029
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = -1 + 4
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 0 + 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001494
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = -1 + 4
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 0 + 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = -1 + 4
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = -1 + 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = -1 + 4
([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
The following 367 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000671The maximin edge-connectivity for choosing a subgraph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001871The number of triconnected components of a graph. St000387The matching number of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001286The annihilation number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000087The number of induced subgraphs. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000383The last part of an integer composition. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St001706The number of closed sets in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000142The number of even parts of a partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000547The number of even non-empty partial sums of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001423The number of distinct cubes in a binary word. St000089The absolute variation of a composition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000377The dinv defect of an integer partition. St000544The cop number of a graph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000913The number of ways to refine the partition into singletons. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001271The competition number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001413Half the length of the longest even length palindromic prefix of a binary word. St001512The minimum rank of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. 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. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001743The discrepancy of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001829The common independence number of a graph. 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. St001917The order of toric promotion on the set of labellings of a graph. St000008The major index of the composition. St000047The number of standard immaculate tableaux of a given shape. St000090The variation of a composition. St000145The Dyson rank of a partition. St000277The number of ribbon shaped standard tableaux. St000299The number of nonisomorphic vertex-induced subtrees. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000392The length of the longest run of ones in a binary word. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000478Another weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000632The jump number of the poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000899The maximal number of repetitions of an integer composition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000982The length of the longest constant subword. St001093The detour number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001716The 1-improper chromatic number of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000184The size of the centralizer of any permutation of given cycle type. St000189The number of elements in the poset. St000228The size of a partition. St000258The burning number of a graph. St000271The chromatic index of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000479The Ramsey number of a graph. St000527The width of the poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000553The number of blocks of a graph. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000708The product of the parts of an integer partition. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000770The major index of an integer partition when read from bottom to top. St000775The multiplicity of the largest eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001279The sum of the parts of an integer partition that are at least two. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001399The distinguishing number of a poset. St001415The length of the longest palindromic prefix of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001530The depth of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001571The Cartan determinant of the integer partition. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001779The order of promotion on the set of linear extensions of a poset. St001828The Euler characteristic of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000180The number of chains of a poset. St000349The number of different adjacency matrices of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001400The total number of Littlewood-Richardson tableaux of given shape. St001664The number of non-isomorphic subposets of a poset. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St000656The number of cuts of a poset. St001117The game chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001615The number of join prime elements of a lattice. St001703The villainy of a graph. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001651The Frankl number of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001720The minimal length of a chain of small intervals in a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001754The number of tolerances of a finite lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001545The second Elser number of a connected graph. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000264The girth of a graph, which is not a tree. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. 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 $eA$. St000699The toughness times the least common multiple of 1,. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition.
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