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Your data matches 515 different statistics following compositions of up to 3 maps.
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Matching statistic: St001325
(load all 65 compositions to match this statistic)
(load all 65 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1 = 2 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph.
A graph is a comparability graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges and $(a,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1 = 2 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001572
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1 = 2 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> 0 = 1 - 1
Description
The minimal number of edges to remove to make a graph bipartite.
Matching statistic: St001793
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1 = 2 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> 0 = 1 - 1
Description
The difference between the clique number and the chromatic number of a graph.
A graph whose clique number and chromatic number coincide are called weakly perfect.
Matching statistic: St000093
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
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: St000273
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000450
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
Description
The number of edges minus the number of vertices plus 2 of a graph.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
Matching statistic: St000482
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
Description
The (zero)-forcing number of a graph.
This is the minimal number of vertices initially coloured black, such that eventually all vertices of the graph are coloured black when using the following rule:
when $u$ is a black vertex of $G$, and exactly one neighbour $v$ of $u$ is white, then colour $v$ black.
Matching statistic: St000544
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
Matching statistic: St000771
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
The following 505 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000671The maximin edge-connectivity for choosing a subgraph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001638The book thickness of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000097The order of the largest clique of the graph. St000183The side length of the Durfee square of an integer partition. St000259The diameter of a connected graph. St000268The number of strongly connected orientations of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000785The number of distinct colouring schemes of a graph. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000917The open packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001073The number of nowhere zero 3-flows of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001432The order dimension of the partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001949The rigidity index of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000089The absolute variation of a composition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000258The burning number of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000469The distinguishing number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000807The sum of the heights of the valleys of the associated bargraph. St000944The 3-degree of an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001315The dissociation number of a graph. St001323The independence gap of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001353The number of prime nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001642The Prague dimension of a graph. St001673The degree of asymmetry of an integer composition. St001674The number of vertices of the largest induced star graph in the graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001871The number of triconnected components of a graph. St000003The number of standard Young tableaux of the partition. 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. St000047The number of standard immaculate tableaux of a given shape. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000667The greatest common divisor of the parts of the partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000820The number of compositions obtained by rotating the composition. St000897The number of different multiplicities of parts of an integer partition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000913The number of ways to refine the partition into singletons. St000920The logarithmic height of a Dyck path. St000948The chromatic discriminant of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is 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. St001093The detour number of a graph. 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 $g$. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001358The largest degree of a regular subgraph of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001571The Cartan determinant of the integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001734The lettericity of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. 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. St000010The length of the partition. St000090The variation of a composition. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000143The largest repeated part of a partition. St000146The Andrews-Garvan crank of a partition. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000160The multiplicity of the smallest part of a partition. St000185The weighted size of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000386The number of factors DDU in a Dyck path. St000439The position of the first down step of a Dyck path. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000636The hull number of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000759The smallest missing part in an integer partition. St000761The number of ascents in an integer composition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001307The number of induced stars on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001484The number of singletons of an integer partition. St001512The minimum rank of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. 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. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001655The general position number of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. 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. 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. St001624The breadth of a lattice. 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. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001763The Hurwitz number of an integer partition. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001623The number of doubly irreducible elements of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001703The villainy of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000455The second largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001625The Möbius invariant of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001833The number of linear intervals in a lattice. 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 $eA$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001545The second Elser number of a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001620The number of sublattices of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000680The Grundy value for Hackendot on posets. St000928The sum of the coefficients of the character polynomial of an integer partition. St000806The semiperimeter of the associated bargraph. St000444The length of the maximal rise of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000369The dinv deficit of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St000477The weight of a partition according to Alladi. St000762The sum of the positions of the weak records of an integer composition. St000997The even-odd crank of an integer partition. St001651The Frankl number of a lattice. St000699The toughness times the least common multiple of 1,. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000068The number of minimal elements in a poset. St001890The maximum magnitude of the Möbius function of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000264The girth of a graph, which is not a tree. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000968We 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 $c_{n−1}$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001163The number of simple modules with dominant dimension at least three 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. 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000069The number of maximal elements of a poset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001248Sum of the even parts of a partition. St001657The number of twos in an integer partition. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000567The sum of the products of all pairs of parts. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001108The 2-dynamic chromatic number of a graph. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001060The distinguishing index of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. 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. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000181The number of connected components of the Hasse diagram for the poset. St000775The multiplicity of the largest eigenvalue in a graph. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001739The number of graphs with the same edge polytope as the given graph. St000449The number of pairs of vertices of a graph with distance 4. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001875The number of simple modules with projective dimension at most 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St000096The number of spanning trees of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001722The number of minimal chains with small intervals between a binary word and the top element.
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