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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000482
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 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: St000776
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St000986
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001570
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001227
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,4),(2,3),(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)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(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)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(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)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000287
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 3 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 1 + 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(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)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(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)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(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)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 4 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 3 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2 + 1
Description
The number of connected components of a graph.
Matching statistic: St001060
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? = 3 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? = 3 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 3 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 1 + 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(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)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(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)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(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)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? = 3 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? = 3 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],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 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 3 + 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 3 = 1 + 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 3 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 3 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(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)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(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)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(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)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(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)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3)],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)],6)
=> ([],1)
=> ([],1)
=> ? = 3 + 2
([(0,1),(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)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(1,2),(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),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 3 + 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(0,1),(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)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ? = 2 + 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,1),(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)],6)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> 3 = 1 + 2
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(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)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(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)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001491
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => 10100 => ? = 3 + 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => ? = 1 + 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => ? = 1 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => 10100 => ? = 3 + 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => 10100 => ? = 3 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 111110 => ? = 2 + 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [3,3] => 100100 => ? = 3 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => 110110 => ? = 2 + 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => 101100 => ? = 3 + 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 111110 => ? = 2 + 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 111110 => ? = 2 + 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 2 + 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => 110110 => ? = 2 + 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => 110101 => ? = 2 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 111110 => ? = 2 + 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => ? = 1 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => 110110 => ? = 2 + 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => ? = 1 + 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,4),(2,3),(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)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,4),(1,5),(2,3),(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)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => ? = 2 + 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => ? = 1 + 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => 111111 => ? = 1 + 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4 = 1 + 3
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001719
Values
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,20),(1,21),(1,22),(1,23),(1,86),(1,87),(1,88),(1,89),(2,25),(2,33),(2,41),(2,47),(2,51),(2,55),(2,86),(2,93),(2,97),(3,24),(3,32),(3,40),(3,46),(3,50),(3,54),(3,86),(3,92),(3,96),(4,26),(4,34),(4,42),(4,44),(4,50),(4,53),(4,87),(4,93),(4,94),(5,27),(5,35),(5,43),(5,45),(5,51),(5,52),(5,87),(5,92),(5,95),(6,28),(6,36),(6,43),(6,44),(6,49),(6,54),(6,88),(6,90),(6,97),(7,29),(7,37),(7,42),(7,45),(7,48),(7,55),(7,88),(7,91),(7,96),(8,31),(8,39),(8,40),(8,47),(8,48),(8,52),(8,89),(8,90),(8,94),(9,30),(9,38),(9,41),(9,46),(9,49),(9,53),(9,89),(9,91),(9,95),(10,15),(10,18),(10,19),(10,21),(10,28),(10,29),(10,30),(10,31),(10,92),(10,93),(11,15),(11,16),(11,17),(11,20),(11,24),(11,25),(11,26),(11,27),(11,90),(11,91),(12,14),(12,17),(12,19),(12,23),(12,34),(12,35),(12,38),(12,39),(12,96),(12,97),(13,14),(13,16),(13,18),(13,22),(13,32),(13,33),(13,36),(13,37),(13,94),(13,95),(14,57),(14,102),(14,103),(14,104),(14,105),(14,138),(15,56),(15,98),(15,99),(15,100),(15,101),(15,138),(16,74),(16,78),(16,79),(16,138),(16,139),(16,140),(17,75),(17,80),(17,81),(17,138),(17,141),(17,142),(18,76),(18,82),(18,83),(18,138),(18,143),(18,144),(19,77),(19,84),(19,85),(19,138),(19,145),(19,146),(20,56),(20,74),(20,75),(20,122),(20,123),(20,151),(21,56),(21,76),(21,77),(21,124),(21,125),(21,152),(22,57),(22,74),(22,76),(22,126),(22,128),(22,153),(23,57),(23,75),(23,77),(23,127),(23,129),(23,154),(24,66),(24,78),(24,98),(24,108),(24,122),(24,141),(25,67),(25,79),(25,99),(25,109),(25,122),(25,142),(26,66),(26,80),(26,99),(26,106),(26,123),(26,139),(27,67),(27,81),(27,98),(27,107),(27,123),(27,140),(28,68),(28,83),(28,101),(28,112),(28,124),(28,146),(29,69),(29,82),(29,100),(29,113),(29,124),(29,145),(30,68),(30,84),(30,100),(30,111),(30,125),(30,143),(31,69),(31,85),(31,101),(31,110),(31,125),(31,144),(32,70),(32,78),(32,102),(32,118),(32,126),(32,143),(33,71),(33,79),(33,103),(33,119),(33,126),(33,144),(34,72),(34,80),(34,105),(34,120),(34,127),(34,146),(35,73),(35,81),(35,104),(35,121),(35,127),(35,145),(36,70),(36,83),(36,103),(36,114),(36,128),(36,139),(37,71),(37,82),(37,102),(37,115),(37,128),(37,140),(38,72),(38,84),(38,104),(38,117),(38,129),(38,141),(39,73),(39,85),(39,105),(39,116),(39,129),(39,142),(40,108),(40,110),(40,116),(40,118),(40,137),(41,109),(41,111),(41,117),(41,119),(41,137),(42,106),(42,113),(42,115),(42,120),(42,136),(43,107),(43,112),(43,114),(43,121),(43,136),(44,59),(44,62),(44,136),(44,139),(44,146),(45,58),(45,63),(45,136),(45,140),(45,145),(46,60),(46,64),(46,137),(46,141),(46,143),(47,61),(47,65),(47,137),(47,142),(47,144),(48,58),(48,61),(48,69),(48,115),(48,116),(48,151),(49,59),(49,60),(49,68),(49,114),(49,117),(49,151),(50,62),(50,64),(50,66),(50,118),(50,120),(50,152),(51,63),(51,65),(51,67),(51,119),(51,121),(51,152),(52,58),(52,65),(52,73),(52,107),(52,110),(52,153),(53,59),(53,64),(53,72),(53,106),(53,111),(53,153),(54,60),(54,62),(54,70),(54,108),(54,112),(54,154),(55,61),(55,63),(55,71),(55,109),(55,113),(55,154),(56,163),(56,164),(56,165),(57,163),(57,166),(57,167),(58,131),(58,161),(58,170),(59,130),(59,162),(59,170),(60,130),(60,159),(60,171),(61,131),(61,160),(61,171),(62,130),(62,155),(62,173),(63,131),(63,156),(63,173),(64,130),(64,157),(64,172),(65,131),(65,158),(65,172),(66,155),(66,157),(66,164),(67,156),(67,158),(67,164),(68,159),(68,162),(68,165),(69,160),(69,161),(69,165),(70,155),(70,159),(70,166),(71,156),(71,160),(71,166),(72,157),(72,162),(72,167),(73,158),(73,161),(73,167),(74,132),(74,163),(74,170),(75,133),(75,163),(75,171),(76,134),(76,163),(76,172),(77,135),(77,163),(77,173),(78,132),(78,155),(78,168),(79,132),(79,156),(79,169),(80,133),(80,157),(80,169),(81,133),(81,158),(81,168),(82,134),(82,160),(82,168),(83,134),(83,159),(83,169),(84,135),(84,162),(84,168),(85,135),(85,161),(85,169),(86,122),(86,126),(86,137),(86,152),(86,154),(87,123),(87,127),(87,136),(87,152),(87,153),(88,124),(88,128),(88,136),(88,151),(88,154),(89,125),(89,129),(89,137),(89,151),(89,153),(90,101),(90,107),(90,108),(90,139),(90,142),(90,151),(91,100),(91,106),(91,109),(91,140),(91,141),(91,151),(92,98),(92,110),(92,112),(92,143),(92,145),(92,152),(93,99),(93,111),(93,113),(93,144),(93,146),(93,152),(94,105),(94,115),(94,118),(94,139),(94,144),(94,153),(95,104),(95,114),(95,119),(95,140),(95,143),(95,153),(96,102),(96,116),(96,120),(96,141),(96,145),(96,154),(97,103),(97,117),(97,121),(97,142),(97,146),(97,154),(98,147),(98,164),(98,168),(99,148),(99,164),(99,169),(100,148),(100,165),(100,168),(101,147),(101,165),(101,169),(102,149),(102,166),(102,168),(103,150),(103,166),(103,169),(104,150),(104,167),(104,168),(105,149),(105,167),(105,169),(106,148),(106,157),(106,170),(107,147),(107,158),(107,170),(108,147),(108,155),(108,171),(109,148),(109,156),(109,171),(110,147),(110,161),(110,172),(111,148),(111,162),(111,172),(112,147),(112,159),(112,173),(113,148),(113,160),(113,173),(114,150),(114,159),(114,170),(115,149),(115,160),(115,170),(116,149),(116,161),(116,171),(117,150),(117,162),(117,171),(118,149),(118,155),(118,172),(119,150),(119,156),(119,172),(120,149),(120,157),(120,173),(121,150),(121,158),(121,173),(122,132),(122,164),(122,171),(123,133),(123,164),(123,170),(124,134),(124,165),(124,173),(125,135),(125,165),(125,172),(126,132),(126,166),(126,172),(127,133),(127,167),(127,173),(128,134),(128,166),(128,170),(129,135),(129,167),(129,171),(130,174),(131,174),(132,174),(133,174),(134,174),(135,174),(136,170),(136,173),(137,171),(137,172),(138,163),(138,168),(138,169),(139,155),(139,169),(139,170),(140,156),(140,168),(140,170),(141,157),(141,168),(141,171),(142,158),(142,169),(142,171),(143,159),(143,168),(143,172),(144,160),(144,169),(144,172),(145,161),(145,168),(145,173),(146,162),(146,169),(146,173),(147,174),(148,174),(149,174),(150,174),(151,165),(151,170),(151,171),(152,164),(152,172),(152,173),(153,167),(153,170),(153,172),(154,166),(154,171),(154,173),(155,174),(156,174),(157,174),(158,174),(159,174),(160,174),(161,174),(162,174),(163,174),(164,174),(165,174),(166,174),(167,174),(168,174),(169,174),(170,174),(171,174),(172,174),(173,174)],175)
=> ? = 3
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,20),(1,29),(1,31),(1,36),(1,40),(1,42),(1,44),(1,81),(1,86),(2,19),(2,28),(2,30),(2,35),(2,39),(2,41),(2,43),(2,81),(2,85),(3,17),(3,26),(3,32),(3,35),(3,37),(3,42),(3,45),(3,82),(3,84),(4,18),(4,27),(4,33),(4,36),(4,38),(4,41),(4,46),(4,82),(4,83),(5,25),(5,37),(5,38),(5,39),(5,40),(5,47),(5,48),(5,79),(5,80),(6,15),(6,16),(6,24),(6,32),(6,33),(6,34),(6,48),(6,85),(6,86),(7,13),(7,14),(7,23),(7,30),(7,31),(7,34),(7,47),(7,83),(7,84),(8,13),(8,17),(8,22),(8,27),(8,43),(8,49),(8,79),(8,86),(9,14),(9,18),(9,21),(9,26),(9,44),(9,49),(9,80),(9,85),(10,15),(10,19),(10,21),(10,29),(10,45),(10,50),(10,79),(10,83),(11,16),(11,20),(11,22),(11,28),(11,46),(11,50),(11,80),(11,84),(12,23),(12,24),(12,25),(12,49),(12,50),(12,81),(12,82),(13,63),(13,91),(13,103),(13,108),(13,127),(14,64),(14,90),(14,102),(14,108),(14,128),(15,65),(15,92),(15,104),(15,109),(15,130),(16,66),(16,93),(16,105),(16,109),(16,129),(17,59),(17,96),(17,101),(17,103),(17,131),(18,60),(18,97),(18,100),(18,102),(18,131),(19,61),(19,94),(19,98),(19,104),(19,132),(20,62),(20,95),(20,99),(20,105),(20,132),(21,73),(21,74),(21,102),(21,104),(21,138),(22,72),(22,75),(22,103),(22,105),(22,138),(23,76),(23,78),(23,108),(23,110),(23,139),(24,77),(24,78),(24,109),(24,111),(24,140),(25,76),(25,77),(25,106),(25,107),(25,138),(26,69),(26,74),(26,114),(26,128),(26,131),(27,68),(27,75),(27,115),(27,127),(27,131),(28,70),(28,72),(28,112),(28,129),(28,132),(29,71),(29,73),(29,113),(29,130),(29,132),(30,53),(30,63),(30,90),(30,98),(30,110),(30,112),(31,54),(31,64),(31,91),(31,99),(31,110),(31,113),(32,56),(32,65),(32,93),(32,101),(32,111),(32,114),(33,55),(33,66),(33,92),(33,100),(33,111),(33,115),(34,67),(34,78),(34,90),(34,91),(34,92),(34,93),(35,51),(35,59),(35,61),(35,112),(35,114),(35,124),(36,52),(36,60),(36,62),(36,113),(36,115),(36,124),(37,51),(37,56),(37,58),(37,96),(37,106),(37,128),(38,52),(38,55),(38,57),(38,97),(38,106),(38,127),(39,51),(39,53),(39,57),(39,94),(39,107),(39,129),(40,52),(40,54),(40,58),(40,95),(40,107),(40,130),(41,57),(41,68),(41,70),(41,98),(41,100),(41,124),(42,58),(42,69),(42,71),(42,99),(42,101),(42,124),(43,59),(43,63),(43,68),(43,72),(43,94),(43,140),(44,60),(44,64),(44,69),(44,73),(44,95),(44,140),(45,61),(45,65),(45,71),(45,74),(45,96),(45,139),(46,62),(46,66),(46,70),(46,75),(46,97),(46,139),(47,53),(47,54),(47,67),(47,76),(47,127),(47,128),(48,55),(48,56),(48,67),(48,77),(48,129),(48,130),(49,108),(49,131),(49,138),(49,140),(50,109),(50,132),(50,138),(50,139),(51,122),(51,133),(51,143),(52,123),(52,133),(52,144),(53,116),(53,134),(53,143),(54,116),(54,135),(54,144),(55,117),(55,136),(55,144),(56,117),(56,137),(56,143),(57,133),(57,134),(57,136),(58,133),(58,135),(58,137),(59,118),(59,122),(59,146),(60,119),(60,123),(60,146),(61,120),(61,122),(61,145),(62,121),(62,123),(62,145),(63,118),(63,134),(63,141),(64,119),(64,135),(64,141),(65,120),(65,137),(65,142),(66,121),(66,136),(66,142),(67,87),(67,143),(67,144),(68,88),(68,134),(68,146),(69,89),(69,135),(69,146),(70,88),(70,136),(70,145),(71,89),(71,137),(71,145),(72,88),(72,118),(72,148),(73,89),(73,119),(73,148),(74,89),(74,120),(74,147),(75,88),(75,121),(75,147),(76,87),(76,116),(76,147),(77,87),(77,117),(77,148),(78,87),(78,141),(78,142),(79,94),(79,96),(79,127),(79,130),(79,138),(80,95),(80,97),(80,128),(80,129),(80,138),(81,107),(81,110),(81,124),(81,132),(81,140),(82,106),(82,111),(82,124),(82,131),(82,139),(83,92),(83,98),(83,102),(83,113),(83,127),(83,139),(84,93),(84,99),(84,103),(84,112),(84,128),(84,139),(85,90),(85,100),(85,104),(85,114),(85,129),(85,140),(86,91),(86,101),(86,105),(86,115),(86,130),(86,140),(87,149),(88,149),(89,149),(90,125),(90,141),(90,143),(91,126),(91,141),(91,144),(92,125),(92,142),(92,144),(93,126),(93,142),(93,143),(94,122),(94,134),(94,148),(95,123),(95,135),(95,148),(96,122),(96,137),(96,147),(97,123),(97,136),(97,147),(98,125),(98,134),(98,145),(99,126),(99,135),(99,145),(100,125),(100,136),(100,146),(101,126),(101,137),(101,146),(102,119),(102,125),(102,147),(103,118),(103,126),(103,147),(104,120),(104,125),(104,148),(105,121),(105,126),(105,148),(106,117),(106,133),(106,147),(107,116),(107,133),(107,148),(108,141),(108,147),(109,142),(109,148),(110,116),(110,141),(110,145),(111,117),(111,142),(111,146),(112,118),(112,143),(112,145),(113,119),(113,144),(113,145),(114,120),(114,143),(114,146),(115,121),(115,144),(115,146),(116,149),(117,149),(118,149),(119,149),(120,149),(121,149),(122,149),(123,149),(124,133),(124,145),(124,146),(125,149),(126,149),(127,134),(127,144),(127,147),(128,135),(128,143),(128,147),(129,136),(129,143),(129,148),(130,137),(130,144),(130,148),(131,146),(131,147),(132,145),(132,148),(133,149),(134,149),(135,149),(136,149),(137,149),(138,147),(138,148),(139,142),(139,145),(139,147),(140,141),(140,146),(140,148),(141,149),(142,149),(143,149),(144,149),(145,149),(146,149),(147,149),(148,149)],150)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,22),(1,27),(1,28),(1,32),(1,35),(1,36),(1,37),(1,40),(1,41),(1,90),(2,21),(2,25),(2,26),(2,31),(2,33),(2,34),(2,37),(2,38),(2,39),(2,89),(3,16),(3,26),(3,30),(3,41),(3,43),(3,45),(3,49),(3,86),(3,92),(4,15),(4,25),(4,29),(4,40),(4,42),(4,44),(4,48),(4,85),(4,92),(5,17),(5,27),(5,29),(5,39),(5,43),(5,46),(5,50),(5,88),(5,91),(6,18),(6,28),(6,30),(6,38),(6,42),(6,47),(6,51),(6,87),(6,91),(7,13),(7,23),(7,24),(7,44),(7,45),(7,46),(7,47),(7,89),(7,90),(8,13),(8,19),(8,20),(8,21),(8,22),(8,59),(8,91),(8,92),(9,14),(9,17),(9,18),(9,19),(9,31),(9,85),(9,86),(9,90),(10,14),(10,15),(10,16),(10,20),(10,32),(10,87),(10,88),(10,89),(11,24),(11,33),(11,35),(11,48),(11,50),(11,59),(11,86),(11,87),(12,23),(12,34),(12,36),(12,49),(12,51),(12,59),(12,85),(12,88),(13,97),(13,98),(13,99),(13,100),(13,103),(14,52),(14,127),(14,128),(14,138),(15,77),(15,96),(15,106),(15,112),(15,127),(16,78),(16,96),(16,107),(16,113),(16,128),(17,75),(17,95),(17,108),(17,111),(17,127),(18,76),(18,95),(18,109),(18,110),(18,128),(19,52),(19,69),(19,95),(19,100),(19,139),(20,52),(20,70),(20,96),(20,99),(20,140),(21,68),(21,69),(21,99),(21,101),(21,104),(21,114),(22,68),(22,70),(22,100),(22,102),(22,105),(22,115),(23,65),(23,67),(23,103),(23,134),(23,136),(24,64),(24,66),(24,103),(24,135),(24,137),(25,55),(25,71),(25,104),(25,106),(25,116),(25,132),(26,56),(26,72),(26,104),(26,107),(26,117),(26,133),(27,58),(27,73),(27,105),(27,108),(27,119),(27,132),(28,57),(28,74),(28,105),(28,109),(28,118),(28,133),(29,60),(29,62),(29,127),(29,129),(29,132),(30,61),(30,63),(30,128),(30,129),(30,133),(31,69),(31,75),(31,76),(31,116),(31,117),(31,138),(32,70),(32,77),(32,78),(32,118),(32,119),(32,138),(33,71),(33,80),(33,84),(33,114),(33,117),(33,135),(34,72),(34,79),(34,83),(34,114),(34,116),(34,134),(35,73),(35,81),(35,84),(35,115),(35,118),(35,137),(36,74),(36,82),(36,83),(36,115),(36,119),(36,136),(37,68),(37,83),(37,84),(37,132),(37,133),(37,138),(38,55),(38,76),(38,79),(38,101),(38,133),(38,135),(39,56),(39,75),(39,80),(39,101),(39,132),(39,134),(40,57),(40,77),(40,81),(40,102),(40,132),(40,136),(41,58),(41,78),(41,82),(41,102),(41,133),(41,137),(42,53),(42,55),(42,57),(42,110),(42,112),(42,129),(43,54),(43,56),(43,58),(43,111),(43,113),(43,129),(44,53),(44,62),(44,66),(44,97),(44,106),(44,136),(45,54),(45,63),(45,67),(45,97),(45,107),(45,137),(46,54),(46,62),(46,64),(46,98),(46,108),(46,134),(47,53),(47,63),(47,65),(47,98),(47,109),(47,135),(48,60),(48,66),(48,71),(48,81),(48,112),(48,139),(49,61),(49,67),(49,72),(49,82),(49,113),(49,139),(50,60),(50,64),(50,73),(50,80),(50,111),(50,140),(51,61),(51,65),(51,74),(51,79),(51,110),(51,140),(52,144),(52,156),(53,143),(53,145),(53,147),(54,143),(54,146),(54,148),(55,121),(55,145),(55,151),(56,122),(56,146),(56,151),(57,123),(57,147),(57,151),(58,124),(58,148),(58,151),(59,103),(59,114),(59,115),(59,139),(59,140),(60,93),(60,141),(60,156),(61,94),(61,142),(61,156),(62,93),(62,143),(62,154),(63,94),(63,143),(63,155),(64,93),(64,148),(64,152),(65,94),(65,147),(65,152),(66,93),(66,145),(66,153),(67,94),(67,146),(67,153),(68,120),(68,144),(68,151),(69,125),(69,144),(69,149),(70,126),(70,144),(70,150),(71,141),(71,145),(71,149),(72,142),(72,146),(72,149),(73,141),(73,148),(73,150),(74,142),(74,147),(74,150),(75,122),(75,125),(75,154),(76,121),(76,125),(76,155),(77,123),(77,126),(77,154),(78,124),(78,126),(78,155),(79,121),(79,142),(79,152),(80,122),(80,141),(80,152),(81,123),(81,141),(81,153),(82,124),(82,142),(82,153),(83,120),(83,142),(83,154),(84,120),(84,141),(84,155),(85,110),(85,116),(85,127),(85,136),(85,139),(86,111),(86,117),(86,128),(86,137),(86,139),(87,112),(87,118),(87,128),(87,135),(87,140),(88,113),(88,119),(88,127),(88,134),(88,140),(89,99),(89,106),(89,107),(89,134),(89,135),(89,138),(90,100),(90,108),(90,109),(90,136),(90,137),(90,138),(91,95),(91,98),(91,101),(91,105),(91,129),(91,140),(92,96),(92,97),(92,102),(92,104),(92,129),(92,139),(93,157),(94,157),(95,125),(95,131),(95,156),(96,126),(96,130),(96,156),(97,130),(97,143),(97,153),(98,131),(98,143),(98,152),(99,130),(99,144),(99,152),(100,131),(100,144),(100,153),(101,125),(101,151),(101,152),(102,126),(102,151),(102,153),(103,152),(103,153),(104,130),(104,149),(104,151),(105,131),(105,150),(105,151),(106,130),(106,145),(106,154),(107,130),(107,146),(107,155),(108,131),(108,148),(108,154),(109,131),(109,147),(109,155),(110,121),(110,147),(110,156),(111,122),(111,148),(111,156),(112,123),(112,145),(112,156),(113,124),(113,146),(113,156),(114,120),(114,149),(114,152),(115,120),(115,150),(115,153),(116,121),(116,149),(116,154),(117,122),(117,149),(117,155),(118,123),(118,150),(118,155),(119,124),(119,150),(119,154),(120,157),(121,157),(122,157),(123,157),(124,157),(125,157),(126,157),(127,154),(127,156),(128,155),(128,156),(129,143),(129,151),(129,156),(130,157),(131,157),(132,141),(132,151),(132,154),(133,142),(133,151),(133,155),(134,146),(134,152),(134,154),(135,145),(135,152),(135,155),(136,147),(136,153),(136,154),(137,148),(137,153),(137,155),(138,144),(138,154),(138,155),(139,149),(139,153),(139,156),(140,150),(140,152),(140,156),(141,157),(142,157),(143,157),(144,157),(145,157),(146,157),(147,157),(148,157),(149,157),(150,157),(151,157),(152,157),(153,157),(154,157),(155,157),(156,157)],158)
=> ? = 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,20),(1,21),(1,32),(1,35),(1,36),(1,41),(1,42),(1,43),(2,15),(2,18),(2,19),(2,31),(2,33),(2,34),(2,39),(2,40),(2,43),(3,23),(3,25),(3,28),(3,38),(3,40),(3,42),(3,71),(3,75),(4,22),(4,24),(4,27),(4,37),(4,39),(4,41),(4,70),(4,75),(5,17),(5,22),(5,23),(5,26),(5,31),(5,32),(5,72),(5,73),(6,13),(6,24),(6,29),(6,33),(6,35),(6,38),(6,72),(6,74),(7,12),(7,25),(7,30),(7,34),(7,36),(7,37),(7,73),(7,74),(8,12),(8,13),(8,14),(8,17),(8,43),(8,70),(8,71),(9,14),(9,15),(9,16),(9,26),(9,44),(9,74),(9,75),(10,18),(10,20),(10,27),(10,30),(10,44),(10,71),(10,72),(11,19),(11,21),(11,28),(11,29),(11,44),(11,70),(11,73),(12,80),(12,88),(12,94),(12,108),(13,80),(13,87),(13,95),(13,109),(14,45),(14,80),(14,83),(14,117),(15,62),(15,81),(15,83),(15,84),(15,92),(16,63),(16,82),(16,83),(16,85),(16,93),(17,45),(17,91),(17,108),(17,109),(18,58),(18,65),(18,84),(18,96),(18,111),(19,59),(19,64),(19,84),(19,97),(19,110),(20,60),(20,67),(20,85),(20,99),(20,111),(21,61),(21,66),(21,85),(21,98),(21,110),(22,46),(22,48),(22,86),(22,89),(22,108),(23,47),(23,49),(23,86),(23,90),(23,109),(24,50),(24,52),(24,87),(24,89),(24,107),(25,51),(25,53),(25,88),(25,90),(25,107),(26,45),(26,62),(26,63),(26,86),(26,118),(27,58),(27,60),(27,68),(27,89),(27,117),(28,59),(28,61),(28,69),(28,90),(28,117),(29,64),(29,66),(29,69),(29,87),(29,118),(30,65),(30,67),(30,68),(30,88),(30,118),(31,46),(31,47),(31,62),(31,91),(31,96),(31,97),(32,48),(32,49),(32,63),(32,91),(32,98),(32,99),(33,50),(33,55),(33,64),(33,92),(33,95),(33,96),(34,51),(34,54),(34,65),(34,92),(34,94),(34,97),(35,52),(35,57),(35,66),(35,93),(35,95),(35,99),(36,53),(36,56),(36,67),(36,93),(36,94),(36,98),(37,54),(37,56),(37,68),(37,107),(37,108),(38,55),(38,57),(38,69),(38,107),(38,109),(39,46),(39,50),(39,54),(39,58),(39,81),(39,110),(40,47),(40,51),(40,55),(40,59),(40,81),(40,111),(41,48),(41,52),(41,56),(41,60),(41,82),(41,110),(42,49),(42,53),(42,57),(42,61),(42,82),(42,111),(43,83),(43,91),(43,94),(43,95),(43,110),(43,111),(44,84),(44,85),(44,117),(44,118),(45,106),(45,124),(46,100),(46,104),(46,121),(47,101),(47,104),(47,122),(48,102),(48,105),(48,121),(49,103),(49,105),(49,122),(50,100),(50,113),(50,115),(51,101),(51,114),(51,115),(52,102),(52,113),(52,116),(53,103),(53,114),(53,116),(54,76),(54,115),(54,121),(55,77),(55,115),(55,122),(56,78),(56,116),(56,121),(57,79),(57,116),(57,122),(58,76),(58,100),(58,123),(59,77),(59,101),(59,123),(60,78),(60,102),(60,123),(61,79),(61,103),(61,123),(62,104),(62,106),(62,119),(63,105),(63,106),(63,120),(64,77),(64,113),(64,119),(65,76),(65,114),(65,119),(66,79),(66,113),(66,120),(67,78),(67,114),(67,120),(68,76),(68,78),(68,124),(69,77),(69,79),(69,124),(70,87),(70,108),(70,110),(70,117),(71,88),(71,109),(71,111),(71,117),(72,89),(72,96),(72,99),(72,109),(72,118),(73,90),(73,97),(73,98),(73,108),(73,118),(74,80),(74,92),(74,93),(74,107),(74,118),(75,81),(75,82),(75,86),(75,107),(75,117),(76,125),(77,125),(78,125),(79,125),(80,112),(80,124),(81,104),(81,115),(81,123),(82,105),(82,116),(82,123),(83,106),(83,112),(83,123),(84,119),(84,123),(85,120),(85,123),(86,104),(86,105),(86,124),(87,113),(87,124),(88,114),(88,124),(89,100),(89,102),(89,124),(90,101),(90,103),(90,124),(91,106),(91,121),(91,122),(92,112),(92,115),(92,119),(93,112),(93,116),(93,120),(94,112),(94,114),(94,121),(95,112),(95,113),(95,122),(96,100),(96,119),(96,122),(97,101),(97,119),(97,121),(98,103),(98,120),(98,121),(99,102),(99,120),(99,122),(100,125),(101,125),(102,125),(103,125),(104,125),(105,125),(106,125),(107,115),(107,116),(107,124),(108,121),(108,124),(109,122),(109,124),(110,113),(110,121),(110,123),(111,114),(111,122),(111,123),(112,125),(113,125),(114,125),(115,125),(116,125),(117,123),(117,124),(118,119),(118,120),(118,124),(119,125),(120,125),(121,125),(122,125),(123,125),(124,125)],126)
=> ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,17),(1,30),(1,31),(1,39),(1,40),(1,41),(1,42),(1,49),(2,14),(2,15),(2,28),(2,29),(2,35),(2,36),(2,37),(2,38),(2,49),(3,19),(3,23),(3,27),(3,34),(3,36),(3,40),(3,74),(3,79),(4,18),(4,22),(4,26),(4,33),(4,35),(4,39),(4,74),(4,78),(5,21),(5,22),(5,24),(5,34),(5,37),(5,41),(5,75),(5,76),(6,20),(6,23),(6,25),(6,33),(6,38),(6,42),(6,75),(6,77),(7,13),(7,20),(7,21),(7,29),(7,31),(7,32),(7,78),(7,79),(8,12),(8,18),(8,19),(8,28),(8,30),(8,32),(8,76),(8,77),(9,14),(9,16),(9,25),(9,26),(9,48),(9,76),(9,79),(10,15),(10,17),(10,24),(10,27),(10,48),(10,77),(10,78),(11,12),(11,13),(11,48),(11,49),(11,74),(11,75),(12,43),(12,84),(12,98),(12,121),(13,43),(13,85),(13,99),(13,122),(14,55),(14,56),(14,86),(14,89),(14,109),(15,54),(15,57),(15,87),(15,88),(15,109),(16,59),(16,60),(16,90),(16,93),(16,109),(17,58),(17,61),(17,91),(17,92),(17,109),(18,62),(18,66),(18,84),(18,94),(18,113),(19,63),(19,67),(19,84),(19,95),(19,114),(20,64),(20,68),(20,85),(20,97),(20,113),(21,65),(21,69),(21,85),(21,96),(21,114),(22,50),(22,52),(22,94),(22,96),(22,110),(23,51),(23,53),(23,95),(23,97),(23,110),(24,54),(24,58),(24,71),(24,96),(24,121),(25,55),(25,59),(25,70),(25,97),(25,121),(26,56),(26,60),(26,70),(26,94),(26,122),(27,57),(27,61),(27,71),(27,95),(27,122),(28,62),(28,63),(28,72),(28,86),(28,87),(28,98),(29,64),(29,65),(29,72),(29,88),(29,89),(29,99),(30,66),(30,67),(30,73),(30,90),(30,91),(30,98),(31,68),(31,69),(31,73),(31,92),(31,93),(31,99),(32,43),(32,72),(32,73),(32,113),(32,114),(33,44),(33,46),(33,70),(33,110),(33,113),(34,45),(34,47),(34,71),(34,110),(34,114),(35,44),(35,50),(35,56),(35,62),(35,88),(35,111),(36,45),(36,51),(36,57),(36,63),(36,89),(36,111),(37,45),(37,50),(37,54),(37,65),(37,86),(37,112),(38,44),(38,51),(38,55),(38,64),(38,87),(38,112),(39,46),(39,52),(39,60),(39,66),(39,92),(39,111),(40,47),(40,53),(40,61),(40,67),(40,93),(40,111),(41,47),(41,52),(41,58),(41,69),(41,90),(41,112),(42,46),(42,53),(42,59),(42,68),(42,91),(42,112),(43,100),(43,126),(44,80),(44,117),(44,123),(45,81),(45,118),(45,123),(46,82),(46,119),(46,123),(47,83),(47,120),(47,123),(48,109),(48,121),(48,122),(49,98),(49,99),(49,109),(49,111),(49,112),(50,101),(50,103),(50,123),(51,102),(51,104),(51,123),(52,105),(52,107),(52,123),(53,106),(53,108),(53,123),(54,81),(54,103),(54,124),(55,80),(55,104),(55,124),(56,80),(56,101),(56,125),(57,81),(57,102),(57,125),(58,83),(58,107),(58,124),(59,82),(59,108),(59,124),(60,82),(60,105),(60,125),(61,83),(61,106),(61,125),(62,101),(62,115),(62,117),(63,102),(63,115),(63,118),(64,104),(64,116),(64,117),(65,103),(65,116),(65,118),(66,105),(66,115),(66,119),(67,106),(67,115),(67,120),(68,108),(68,116),(68,119),(69,107),(69,116),(69,120),(70,80),(70,82),(70,126),(71,81),(71,83),(71,126),(72,100),(72,117),(72,118),(73,100),(73,119),(73,120),(74,84),(74,110),(74,111),(74,122),(75,85),(75,110),(75,112),(75,121),(76,86),(76,90),(76,94),(76,114),(76,121),(77,87),(77,91),(77,95),(77,113),(77,121),(78,88),(78,92),(78,96),(78,113),(78,122),(79,89),(79,93),(79,97),(79,114),(79,122),(80,127),(81,127),(82,127),(83,127),(84,115),(84,126),(85,116),(85,126),(86,101),(86,118),(86,124),(87,102),(87,117),(87,124),(88,103),(88,117),(88,125),(89,104),(89,118),(89,125),(90,105),(90,120),(90,124),(91,106),(91,119),(91,124),(92,107),(92,119),(92,125),(93,108),(93,120),(93,125),(94,101),(94,105),(94,126),(95,102),(95,106),(95,126),(96,103),(96,107),(96,126),(97,104),(97,108),(97,126),(98,100),(98,115),(98,124),(99,100),(99,116),(99,125),(100,127),(101,127),(102,127),(103,127),(104,127),(105,127),(106,127),(107,127),(108,127),(109,124),(109,125),(110,123),(110,126),(111,115),(111,123),(111,125),(112,116),(112,123),(112,124),(113,117),(113,119),(113,126),(114,118),(114,120),(114,126),(115,127),(116,127),(117,127),(118,127),(119,127),(120,127),(121,124),(121,126),(122,125),(122,126),(123,127),(124,127),(125,127),(126,127)],128)
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,14),(1,18),(1,22),(1,28),(1,30),(1,36),(1,37),(1,66),(2,13),(2,17),(2,21),(2,27),(2,29),(2,34),(2,35),(2,66),(3,12),(3,20),(3,24),(3,26),(3,32),(3,35),(3,37),(3,67),(4,11),(4,19),(4,23),(4,25),(4,31),(4,34),(4,36),(4,67),(5,16),(5,19),(5,20),(5,27),(5,28),(5,39),(5,40),(6,15),(6,17),(6,18),(6,25),(6,26),(6,38),(6,40),(7,29),(7,30),(7,31),(7,32),(7,33),(7,40),(7,41),(8,21),(8,22),(8,23),(8,24),(8,33),(8,38),(8,39),(9,11),(9,12),(9,15),(9,39),(9,41),(9,66),(10,13),(10,14),(10,16),(10,38),(10,41),(10,67),(11,42),(11,74),(11,80),(11,94),(12,43),(12,75),(12,81),(12,94),(13,44),(13,76),(13,78),(13,95),(14,45),(14,77),(14,79),(14,95),(15,42),(15,43),(15,72),(15,98),(16,44),(16,45),(16,73),(16,98),(17,46),(17,47),(17,72),(17,78),(17,82),(18,48),(18,49),(18,72),(18,79),(18,83),(19,50),(19,52),(19,73),(19,80),(19,84),(20,51),(20,53),(20,73),(20,81),(20,85),(21,54),(21,55),(21,64),(21,78),(21,97),(22,56),(22,57),(22,65),(22,79),(22,97),(23,54),(23,56),(23,62),(23,80),(23,96),(24,55),(24,57),(24,63),(24,81),(24,96),(25,42),(25,46),(25,48),(25,84),(25,96),(26,43),(26,47),(26,49),(26,85),(26,96),(27,44),(27,50),(27,51),(27,82),(27,97),(28,45),(28,52),(28,53),(28,83),(28,97),(29,58),(29,59),(29,64),(29,82),(29,95),(30,60),(30,61),(30,65),(30,83),(30,95),(31,58),(31,60),(31,62),(31,84),(31,94),(32,59),(32,61),(32,63),(32,85),(32,94),(33,62),(33,63),(33,64),(33,65),(33,98),(34,46),(34,50),(34,54),(34,58),(34,74),(34,76),(35,47),(35,51),(35,55),(35,59),(35,75),(35,76),(36,48),(36,52),(36,56),(36,60),(36,74),(36,77),(37,49),(37,53),(37,57),(37,61),(37,75),(37,77),(38,78),(38,79),(38,96),(38,98),(39,80),(39,81),(39,97),(39,98),(40,82),(40,83),(40,84),(40,85),(40,98),(41,94),(41,95),(41,98),(42,90),(42,104),(43,91),(43,104),(44,92),(44,105),(45,93),(45,105),(46,86),(46,90),(46,99),(47,87),(47,91),(47,99),(48,88),(48,90),(48,100),(49,89),(49,91),(49,100),(50,86),(50,92),(50,101),(51,87),(51,92),(51,102),(52,88),(52,93),(52,101),(53,89),(53,93),(53,102),(54,68),(54,99),(54,101),(55,69),(55,99),(55,102),(56,70),(56,100),(56,101),(57,71),(57,100),(57,102),(58,68),(58,86),(58,103),(59,69),(59,87),(59,103),(60,70),(60,88),(60,103),(61,71),(61,89),(61,103),(62,68),(62,70),(62,104),(63,69),(63,71),(63,104),(64,68),(64,69),(64,105),(65,70),(65,71),(65,105),(66,72),(66,74),(66,75),(66,95),(66,97),(67,73),(67,76),(67,77),(67,94),(67,96),(68,106),(69,106),(70,106),(71,106),(72,90),(72,91),(72,105),(73,92),(73,93),(73,104),(74,90),(74,101),(74,103),(75,91),(75,102),(75,103),(76,92),(76,99),(76,103),(77,93),(77,100),(77,103),(78,99),(78,105),(79,100),(79,105),(80,101),(80,104),(81,102),(81,104),(82,86),(82,87),(82,105),(83,88),(83,89),(83,105),(84,86),(84,88),(84,104),(85,87),(85,89),(85,104),(86,106),(87,106),(88,106),(89,106),(90,106),(91,106),(92,106),(93,106),(94,103),(94,104),(95,103),(95,105),(96,99),(96,100),(96,104),(97,101),(97,102),(97,105),(98,104),(98,105),(99,106),(100,106),(101,106),(102,106),(103,106),(104,106),(105,106)],107)
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,20),(1,21),(1,22),(1,23),(1,36),(1,37),(1,67),(2,12),(2,16),(2,17),(2,18),(2,19),(2,34),(2,35),(2,67),(3,15),(3,28),(3,29),(3,30),(3,31),(3,35),(3,37),(3,66),(4,14),(4,24),(4,25),(4,26),(4,27),(4,34),(4,36),(4,66),(5,17),(5,21),(5,25),(5,29),(5,33),(5,39),(5,41),(6,16),(6,20),(6,24),(6,28),(6,33),(6,38),(6,40),(7,18),(7,23),(7,27),(7,30),(7,32),(7,38),(7,41),(8,19),(8,22),(8,26),(8,31),(8,32),(8,39),(8,40),(9,11),(9,14),(9,15),(9,40),(9,41),(9,67),(10,11),(10,12),(10,13),(10,38),(10,39),(10,66),(11,76),(11,77),(11,95),(12,76),(12,78),(12,82),(12,83),(13,76),(13,79),(13,84),(13,85),(14,77),(14,80),(14,86),(14,87),(15,77),(15,81),(15,88),(15,89),(16,42),(16,46),(16,58),(16,82),(16,93),(17,43),(17,47),(17,58),(17,83),(17,94),(18,44),(18,48),(18,59),(18,82),(18,94),(19,45),(19,49),(19,59),(19,83),(19,93),(20,50),(20,54),(20,60),(20,84),(20,93),(21,51),(21,55),(21,60),(21,85),(21,94),(22,52),(22,56),(22,61),(22,85),(22,93),(23,53),(23,57),(23,61),(23,84),(23,94),(24,42),(24,50),(24,62),(24,86),(24,91),(25,43),(25,51),(25,62),(25,87),(25,92),(26,45),(26,52),(26,63),(26,86),(26,92),(27,44),(27,53),(27,63),(27,87),(27,91),(28,46),(28,54),(28,64),(28,88),(28,91),(29,47),(29,55),(29,64),(29,89),(29,92),(30,48),(30,57),(30,65),(30,89),(30,91),(31,49),(31,56),(31,65),(31,88),(31,92),(32,59),(32,61),(32,63),(32,65),(32,95),(33,58),(33,60),(33,62),(33,64),(33,95),(34,42),(34,43),(34,44),(34,45),(34,78),(34,80),(35,46),(35,47),(35,48),(35,49),(35,78),(35,81),(36,50),(36,51),(36,52),(36,53),(36,79),(36,80),(37,54),(37,55),(37,56),(37,57),(37,79),(37,81),(38,82),(38,84),(38,91),(38,95),(39,83),(39,85),(39,92),(39,95),(40,86),(40,88),(40,93),(40,95),(41,87),(41,89),(41,94),(41,95),(42,68),(42,96),(42,100),(43,68),(43,97),(43,101),(44,69),(44,96),(44,101),(45,69),(45,97),(45,100),(46,70),(46,96),(46,102),(47,70),(47,97),(47,103),(48,71),(48,96),(48,103),(49,71),(49,97),(49,102),(50,72),(50,98),(50,100),(51,72),(51,99),(51,101),(52,73),(52,99),(52,100),(53,73),(53,98),(53,101),(54,74),(54,98),(54,102),(55,74),(55,99),(55,103),(56,75),(56,99),(56,102),(57,75),(57,98),(57,103),(58,68),(58,70),(58,104),(59,69),(59,71),(59,104),(60,72),(60,74),(60,104),(61,73),(61,75),(61,104),(62,68),(62,72),(62,105),(63,69),(63,73),(63,105),(64,70),(64,74),(64,105),(65,71),(65,75),(65,105),(66,77),(66,78),(66,79),(66,91),(66,92),(67,76),(67,80),(67,81),(67,93),(67,94),(68,106),(69,106),(70,106),(71,106),(72,106),(73,106),(74,106),(75,106),(76,90),(76,104),(77,90),(77,105),(78,90),(78,96),(78,97),(79,90),(79,98),(79,99),(80,90),(80,100),(80,101),(81,90),(81,102),(81,103),(82,96),(82,104),(83,97),(83,104),(84,98),(84,104),(85,99),(85,104),(86,100),(86,105),(87,101),(87,105),(88,102),(88,105),(89,103),(89,105),(90,106),(91,96),(91,98),(91,105),(92,97),(92,99),(92,105),(93,100),(93,102),(93,104),(94,101),(94,103),(94,104),(95,104),(95,105),(96,106),(97,106),(98,106),(99,106),(100,106),(101,106),(102,106),(103,106),(104,106),(105,106)],107)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(1,17),(1,24),(1,38),(1,40),(2,10),(2,16),(2,24),(2,37),(2,39),(3,12),(3,18),(3,23),(3,37),(3,40),(4,13),(4,19),(4,23),(4,38),(4,39),(5,15),(5,21),(5,22),(5,39),(5,40),(6,14),(6,20),(6,22),(6,37),(6,38),(7,9),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,31),(9,32),(9,33),(9,34),(9,35),(9,36),(10,25),(10,31),(10,41),(10,43),(11,25),(11,32),(11,42),(11,44),(12,26),(12,33),(12,41),(12,44),(13,26),(13,34),(13,42),(13,43),(14,27),(14,35),(14,41),(14,42),(15,27),(15,36),(15,43),(15,44),(16,28),(16,31),(16,45),(16,47),(17,28),(17,32),(17,46),(17,48),(18,29),(18,33),(18,45),(18,48),(19,29),(19,34),(19,46),(19,47),(20,30),(20,35),(20,45),(20,46),(21,30),(21,36),(21,47),(21,48),(22,27),(22,30),(22,56),(23,26),(23,29),(23,56),(24,25),(24,28),(24,56),(25,49),(25,57),(26,50),(26,57),(27,51),(27,57),(28,49),(28,58),(29,50),(29,58),(30,51),(30,58),(31,49),(31,52),(31,54),(32,49),(32,53),(32,55),(33,50),(33,52),(33,55),(34,50),(34,53),(34,54),(35,51),(35,52),(35,53),(36,51),(36,54),(36,55),(37,41),(37,45),(37,56),(38,42),(38,46),(38,56),(39,43),(39,47),(39,56),(40,44),(40,48),(40,56),(41,52),(41,57),(42,53),(42,57),(43,54),(43,57),(44,55),(44,57),(45,52),(45,58),(46,53),(46,58),(47,54),(47,58),(48,55),(48,58),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,59),(56,57),(56,58),(57,59),(58,59)],60)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,14),(1,22),(1,23),(1,32),(1,36),(1,40),(1,48),(1,90),(1,94),(2,13),(2,21),(2,23),(2,31),(2,35),(2,39),(2,47),(2,89),(2,93),(3,16),(3,19),(3,24),(3,34),(3,35),(3,41),(3,49),(3,90),(3,91),(4,15),(4,20),(4,24),(4,33),(4,36),(4,42),(4,50),(4,89),(4,92),(5,15),(5,19),(5,25),(5,31),(5,38),(5,43),(5,51),(5,87),(5,94),(6,16),(6,20),(6,25),(6,32),(6,37),(6,44),(6,52),(6,88),(6,93),(7,13),(7,22),(7,26),(7,33),(7,37),(7,45),(7,53),(7,87),(7,91),(8,14),(8,21),(8,26),(8,34),(8,38),(8,46),(8,54),(8,88),(8,92),(9,18),(9,29),(9,30),(9,43),(9,44),(9,45),(9,46),(9,89),(9,90),(10,18),(10,27),(10,28),(10,39),(10,40),(10,41),(10,42),(10,87),(10,88),(11,17),(11,28),(11,30),(11,49),(11,50),(11,53),(11,54),(11,93),(11,94),(12,17),(12,27),(12,29),(12,47),(12,48),(12,51),(12,52),(12,91),(12,92),(13,95),(13,101),(13,103),(13,109),(13,127),(14,96),(14,102),(14,104),(14,110),(14,127),(15,97),(15,99),(15,106),(15,108),(15,126),(16,98),(16,100),(16,105),(16,107),(16,126),(17,115),(17,116),(17,117),(17,118),(17,125),(18,111),(18,112),(18,113),(18,114),(18,125),(19,63),(19,68),(19,126),(19,132),(19,139),(20,64),(20,67),(20,126),(20,133),(20,138),(21,65),(21,69),(21,127),(21,134),(21,136),(22,66),(22,70),(22,127),(22,135),(22,137),(23,55),(23,59),(23,127),(23,129),(23,131),(24,56),(24,60),(24,126),(24,129),(24,130),(25,57),(25,61),(25,126),(25,128),(25,131),(26,58),(26,62),(26,127),(26,128),(26,130),(27,71),(27,72),(27,125),(27,132),(27,133),(28,73),(28,74),(28,125),(28,134),(28,135),(29,75),(29,76),(29,125),(29,136),(29,137),(30,77),(30,78),(30,125),(30,138),(30,139),(31,63),(31,69),(31,83),(31,95),(31,99),(31,131),(32,64),(32,70),(32,84),(32,96),(32,100),(32,131),(33,66),(33,67),(33,85),(33,97),(33,101),(33,130),(34,65),(34,68),(34,86),(34,98),(34,102),(34,130),(35,63),(35,65),(35,79),(35,103),(35,105),(35,129),(36,64),(36,66),(36,80),(36,104),(36,106),(36,129),(37,67),(37,70),(37,81),(37,107),(37,109),(37,128),(38,68),(38,69),(38,82),(38,108),(38,110),(38,128),(39,55),(39,71),(39,79),(39,95),(39,111),(39,134),(40,55),(40,72),(40,80),(40,96),(40,112),(40,135),(41,56),(41,73),(41,79),(41,98),(41,112),(41,132),(42,56),(42,74),(42,80),(42,97),(42,111),(42,133),(43,57),(43,75),(43,82),(43,99),(43,113),(43,139),(44,57),(44,76),(44,81),(44,100),(44,114),(44,138),(45,58),(45,77),(45,81),(45,101),(45,113),(45,137),(46,58),(46,78),(46,82),(46,102),(46,114),(46,136),(47,59),(47,71),(47,83),(47,103),(47,115),(47,136),(48,59),(48,72),(48,84),(48,104),(48,116),(48,137),(49,60),(49,73),(49,86),(49,105),(49,117),(49,139),(50,60),(50,74),(50,85),(50,106),(50,118),(50,138),(51,61),(51,75),(51,83),(51,108),(51,116),(51,132),(52,61),(52,76),(52,84),(52,107),(52,115),(52,133),(53,62),(53,77),(53,85),(53,109),(53,117),(53,135),(54,62),(54,78),(54,86),(54,110),(54,118),(54,134),(55,121),(55,144),(55,157),(56,122),(56,144),(56,156),(57,123),(57,145),(57,159),(58,124),(58,145),(58,158),(59,121),(59,146),(59,158),(60,122),(60,147),(60,159),(61,123),(61,146),(61,156),(62,124),(62,147),(62,157),(63,119),(63,148),(63,159),(64,120),(64,149),(64,159),(65,119),(65,150),(65,158),(66,120),(66,151),(66,158),(67,120),(67,154),(67,156),(68,119),(68,155),(68,156),(69,119),(69,152),(69,157),(70,120),(70,153),(70,157),(71,121),(71,148),(71,160),(72,121),(72,149),(72,161),(73,122),(73,150),(73,161),(74,122),(74,151),(74,160),(75,123),(75,152),(75,161),(76,123),(76,153),(76,160),(77,124),(77,154),(77,161),(78,124),(78,155),(78,160),(79,144),(79,148),(79,150),(80,144),(80,149),(80,151),(81,145),(81,153),(81,154),(82,145),(82,152),(82,155),(83,146),(83,148),(83,152),(84,146),(84,149),(84,153),(85,147),(85,151),(85,154),(86,147),(86,150),(86,155),(87,95),(87,97),(87,113),(87,128),(87,132),(87,135),(88,96),(88,98),(88,114),(88,128),(88,133),(88,134),(89,99),(89,101),(89,111),(89,129),(89,136),(89,138),(90,100),(90,102),(90,112),(90,129),(90,137),(90,139),(91,103),(91,107),(91,117),(91,130),(91,132),(91,137),(92,104),(92,108),(92,118),(92,130),(92,133),(92,136),(93,105),(93,109),(93,115),(93,131),(93,134),(93,138),(94,106),(94,110),(94,116),(94,131),(94,135),(94,139),(95,140),(95,148),(95,157),(96,141),(96,149),(96,157),(97,140),(97,151),(97,156),(98,141),(98,150),(98,156),(99,140),(99,152),(99,159),(100,141),(100,153),(100,159),(101,140),(101,154),(101,158),(102,141),(102,155),(102,158),(103,142),(103,148),(103,158),(104,143),(104,149),(104,158),(105,142),(105,150),(105,159),(106,143),(106,151),(106,159),(107,142),(107,153),(107,156),(108,143),(108,152),(108,156),(109,142),(109,154),(109,157),(110,143),(110,155),(110,157),(111,140),(111,144),(111,160),(112,141),(112,144),(112,161),(113,140),(113,145),(113,161),(114,141),(114,145),(114,160),(115,142),(115,146),(115,160),(116,143),(116,146),(116,161),(117,142),(117,147),(117,161),(118,143),(118,147),(118,160),(119,162),(120,162),(121,162),(122,162),(123,162),(124,162),(125,160),(125,161),(126,156),(126,159),(127,157),(127,158),(128,145),(128,156),(128,157),(129,144),(129,158),(129,159),(130,147),(130,156),(130,158),(131,146),(131,157),(131,159),(132,148),(132,156),(132,161),(133,149),(133,156),(133,160),(134,150),(134,157),(134,160),(135,151),(135,157),(135,161),(136,152),(136,158),(136,160),(137,153),(137,158),(137,161),(138,154),(138,159),(138,160),(139,155),(139,159),(139,161),(140,162),(141,162),(142,162),(143,162),(144,162),(145,162),(146,162),(147,162),(148,162),(149,162),(150,162),(151,162),(152,162),(153,162),(154,162),(155,162),(156,162),(157,162),(158,162),(159,162),(160,162),(161,162)],163)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,30),(1,31),(1,32),(1,33),(1,47),(1,48),(1,84),(1,85),(2,21),(2,25),(2,35),(2,38),(2,42),(2,46),(2,83),(2,85),(3,20),(3,24),(3,34),(3,38),(3,41),(3,45),(3,82),(3,84),(4,19),(4,22),(4,37),(4,39),(4,43),(4,45),(4,80),(4,85),(5,18),(5,23),(5,36),(5,39),(5,44),(5,46),(5,81),(5,84),(6,28),(6,29),(6,40),(6,43),(6,44),(6,48),(6,82),(6,83),(7,26),(7,27),(7,40),(7,41),(7,42),(7,47),(7,80),(7,81),(8,12),(8,15),(8,16),(8,18),(8,22),(8,26),(8,30),(8,34),(8,83),(9,12),(9,14),(9,17),(9,19),(9,23),(9,27),(9,31),(9,35),(9,82),(10,13),(10,14),(10,16),(10,20),(10,25),(10,28),(10,32),(10,36),(10,80),(11,13),(11,15),(11,17),(11,21),(11,24),(11,29),(11,33),(11,37),(11,81),(12,49),(12,54),(12,117),(12,119),(12,121),(13,50),(13,55),(13,118),(13,119),(13,120),(14,56),(14,59),(14,64),(14,86),(14,88),(14,119),(15,57),(15,58),(15,65),(15,87),(15,89),(15,119),(16,70),(16,76),(16,78),(16,98),(16,101),(16,119),(17,71),(17,77),(17,79),(17,99),(17,100),(17,119),(18,76),(18,87),(18,97),(18,102),(18,117),(19,77),(19,86),(19,96),(19,103),(19,117),(20,78),(20,88),(20,94),(20,104),(20,118),(21,79),(21,89),(21,95),(21,105),(21,118),(22,57),(22,66),(22,98),(22,117),(22,125),(23,56),(23,67),(23,99),(23,117),(23,124),(24,58),(24,68),(24,100),(24,118),(24,122),(25,59),(25,69),(25,101),(25,118),(25,123),(26,54),(26,60),(26,72),(26,87),(26,91),(26,98),(27,54),(27,61),(27,73),(27,86),(27,90),(27,99),(28,55),(28,62),(28,74),(28,88),(28,92),(28,101),(29,55),(29,63),(29,75),(29,89),(29,93),(29,100),(30,49),(30,60),(30,65),(30,70),(30,102),(30,125),(31,49),(31,61),(31,64),(31,71),(31,103),(31,124),(32,50),(32,62),(32,64),(32,70),(32,104),(32,123),(33,50),(33,63),(33,65),(33,71),(33,105),(33,122),(34,58),(34,66),(34,72),(34,78),(34,102),(34,121),(35,59),(35,67),(35,73),(35,79),(35,103),(35,121),(36,56),(36,69),(36,74),(36,76),(36,104),(36,120),(37,57),(37,68),(37,75),(37,77),(37,105),(37,120),(38,52),(38,116),(38,118),(38,121),(39,53),(39,116),(39,117),(39,120),(40,51),(40,90),(40,91),(40,92),(40,93),(41,52),(41,72),(41,90),(41,94),(41,122),(42,52),(42,73),(42,91),(42,95),(42,123),(43,53),(43,75),(43,92),(43,96),(43,125),(44,53),(44,74),(44,93),(44,97),(44,124),(45,66),(45,68),(45,94),(45,96),(45,116),(46,67),(46,69),(46,95),(46,97),(46,116),(47,51),(47,60),(47,61),(47,122),(47,123),(48,51),(48,62),(48,63),(48,124),(48,125),(49,106),(49,134),(49,140),(50,107),(50,134),(50,139),(51,135),(51,136),(52,128),(52,139),(53,129),(53,140),(54,106),(54,128),(54,137),(55,107),(55,129),(55,138),(56,109),(56,132),(56,137),(57,108),(57,133),(57,137),(58,108),(58,130),(58,138),(59,109),(59,131),(59,138),(60,106),(60,130),(60,136),(61,106),(61,131),(61,135),(62,107),(62,132),(62,136),(63,107),(63,133),(63,135),(64,131),(64,132),(64,134),(65,130),(65,133),(65,134),(66,108),(66,112),(66,140),(67,109),(67,113),(67,140),(68,108),(68,114),(68,139),(69,109),(69,115),(69,139),(70,110),(70,134),(70,136),(71,111),(71,134),(71,135),(72,112),(72,128),(72,130),(73,113),(73,128),(73,131),(74,115),(74,129),(74,132),(75,114),(75,129),(75,133),(76,110),(76,115),(76,137),(77,111),(77,114),(77,137),(78,110),(78,112),(78,138),(79,111),(79,113),(79,138),(80,86),(80,92),(80,94),(80,98),(80,120),(80,123),(81,87),(81,93),(81,95),(81,99),(81,120),(81,122),(82,88),(82,90),(82,96),(82,100),(82,121),(82,124),(83,89),(83,91),(83,97),(83,101),(83,121),(83,125),(84,102),(84,104),(84,116),(84,122),(84,124),(85,103),(85,105),(85,116),(85,123),(85,125),(86,126),(86,131),(86,137),(87,127),(87,130),(87,137),(88,126),(88,132),(88,138),(89,127),(89,133),(89,138),(90,126),(90,128),(90,135),(91,127),(91,128),(91,136),(92,126),(92,129),(92,136),(93,127),(93,129),(93,135),(94,112),(94,126),(94,139),(95,113),(95,127),(95,139),(96,114),(96,126),(96,140),(97,115),(97,127),(97,140),(98,112),(98,136),(98,137),(99,113),(99,135),(99,137),(100,114),(100,135),(100,138),(101,115),(101,136),(101,138),(102,110),(102,130),(102,140),(103,111),(103,131),(103,140),(104,110),(104,132),(104,139),(105,111),(105,133),(105,139),(106,141),(107,141),(108,141),(109,141),(110,141),(111,141),(112,141),(113,141),(114,141),(115,141),(116,139),(116,140),(117,137),(117,140),(118,138),(118,139),(119,134),(119,137),(119,138),(120,129),(120,137),(120,139),(121,128),(121,138),(121,140),(122,130),(122,135),(122,139),(123,131),(123,136),(123,139),(124,132),(124,135),(124,140),(125,133),(125,136),(125,140),(126,141),(127,141),(128,141),(129,141),(130,141),(131,141),(132,141),(133,141),(134,141),(135,141),(136,141),(137,141),(138,141),(139,141),(140,141)],142)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(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),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,15),(1,21),(1,29),(1,30),(1,35),(1,36),(1,44),(1,45),(1,93),(2,14),(2,20),(2,26),(2,28),(2,32),(2,34),(2,43),(2,45),(2,92),(3,13),(3,19),(3,25),(3,27),(3,31),(3,33),(3,43),(3,44),(3,91),(4,16),(4,19),(4,28),(4,29),(4,37),(4,39),(4,46),(4,48),(4,90),(5,17),(5,20),(5,27),(5,30),(5,38),(5,40),(5,47),(5,48),(5,89),(6,18),(6,21),(6,25),(6,26),(6,41),(6,42),(6,46),(6,47),(6,88),(7,22),(7,23),(7,33),(7,34),(7,37),(7,38),(7,88),(7,93),(8,22),(8,24),(8,31),(8,35),(8,39),(8,41),(8,89),(8,92),(9,23),(9,24),(9,32),(9,36),(9,40),(9,42),(9,90),(9,91),(10,12),(10,16),(10,17),(10,18),(10,91),(10,92),(10,93),(11,12),(11,13),(11,14),(11,15),(11,88),(11,89),(11,90),(12,120),(12,121),(12,122),(13,52),(13,53),(13,99),(13,101),(13,120),(14,52),(14,54),(14,100),(14,102),(14,121),(15,53),(15,54),(15,103),(15,104),(15,122),(16,55),(16,57),(16,108),(16,109),(16,120),(17,56),(17,57),(17,107),(17,110),(17,121),(18,55),(18,56),(18,105),(18,106),(18,122),(19,64),(19,66),(19,117),(19,118),(19,120),(20,65),(20,67),(20,117),(20,119),(20,121),(21,68),(21,69),(21,118),(21,119),(21,122),(22,51),(22,70),(22,73),(22,124),(22,128),(23,51),(23,71),(23,74),(23,123),(23,127),(24,51),(24,72),(24,75),(24,125),(24,126),(25,58),(25,61),(25,76),(25,101),(25,105),(25,118),(26,58),(26,62),(26,77),(26,102),(26,106),(26,119),(27,59),(27,61),(27,78),(27,99),(27,107),(27,117),(28,60),(28,62),(28,79),(28,100),(28,108),(28,117),(29,60),(29,63),(29,80),(29,104),(29,109),(29,118),(30,59),(30,63),(30,81),(30,103),(30,110),(30,119),(31,66),(31,70),(31,76),(31,83),(31,99),(31,126),(32,67),(32,71),(32,77),(32,84),(32,100),(32,126),(33,64),(33,70),(33,78),(33,82),(33,101),(33,127),(34,65),(34,71),(34,79),(34,82),(34,102),(34,128),(35,69),(35,72),(35,80),(35,83),(35,103),(35,128),(36,68),(36,72),(36,81),(36,84),(36,104),(36,127),(37,64),(37,73),(37,79),(37,87),(37,109),(37,123),(38,65),(38,74),(38,78),(38,87),(38,110),(38,124),(39,66),(39,73),(39,80),(39,85),(39,108),(39,125),(40,67),(40,74),(40,81),(40,86),(40,107),(40,125),(41,69),(41,75),(41,76),(41,85),(41,106),(41,124),(42,68),(42,75),(42,77),(42,86),(42,105),(42,123),(43,49),(43,52),(43,58),(43,82),(43,117),(43,126),(44,49),(44,53),(44,59),(44,83),(44,118),(44,127),(45,49),(45,54),(45,60),(45,84),(45,119),(45,128),(46,50),(46,55),(46,62),(46,85),(46,118),(46,123),(47,50),(47,56),(47,61),(47,86),(47,119),(47,124),(48,50),(48,57),(48,63),(48,87),(48,117),(48,125),(49,97),(49,139),(49,140),(50,98),(50,138),(50,140),(51,138),(51,139),(52,97),(52,113),(52,143),(53,97),(53,111),(53,141),(54,97),(54,112),(54,142),(55,98),(55,115),(55,141),(56,98),(56,114),(56,142),(57,98),(57,116),(57,143),(58,113),(58,137),(58,140),(59,111),(59,135),(59,140),(60,112),(60,136),(60,140),(61,114),(61,132),(61,140),(62,115),(62,133),(62,140),(63,116),(63,134),(63,140),(64,94),(64,129),(64,141),(65,95),(65,129),(65,142),(66,94),(66,130),(66,143),(67,95),(67,131),(67,143),(68,96),(68,131),(68,141),(69,96),(69,130),(69,142),(70,94),(70,132),(70,139),(71,95),(71,133),(71,139),(72,96),(72,134),(72,139),(73,94),(73,136),(73,138),(74,95),(74,135),(74,138),(75,96),(75,137),(75,138),(76,130),(76,132),(76,137),(77,131),(77,133),(77,137),(78,129),(78,132),(78,135),(79,129),(79,133),(79,136),(80,130),(80,134),(80,136),(81,131),(81,134),(81,135),(82,113),(82,129),(82,139),(83,111),(83,130),(83,139),(84,112),(84,131),(84,139),(85,115),(85,130),(85,138),(86,114),(86,131),(86,138),(87,116),(87,129),(87,138),(88,101),(88,102),(88,122),(88,123),(88,124),(89,99),(89,103),(89,121),(89,124),(89,125),(90,100),(90,104),(90,120),(90,123),(90,125),(91,105),(91,107),(91,120),(91,126),(91,127),(92,106),(92,108),(92,121),(92,126),(92,128),(93,109),(93,110),(93,122),(93,127),(93,128),(94,144),(95,144),(96,144),(97,144),(98,144),(99,111),(99,132),(99,143),(100,112),(100,133),(100,143),(101,113),(101,132),(101,141),(102,113),(102,133),(102,142),(103,111),(103,134),(103,142),(104,112),(104,134),(104,141),(105,114),(105,137),(105,141),(106,115),(106,137),(106,142),(107,114),(107,135),(107,143),(108,115),(108,136),(108,143),(109,116),(109,136),(109,141),(110,116),(110,135),(110,142),(111,144),(112,144),(113,144),(114,144),(115,144),(116,144),(117,129),(117,140),(117,143),(118,130),(118,140),(118,141),(119,131),(119,140),(119,142),(120,141),(120,143),(121,142),(121,143),(122,141),(122,142),(123,133),(123,138),(123,141),(124,132),(124,138),(124,142),(125,134),(125,138),(125,143),(126,137),(126,139),(126,143),(127,135),(127,139),(127,141),(128,136),(128,139),(128,142),(129,144),(130,144),(131,144),(132,144),(133,144),(134,144),(135,144),(136,144),(137,144),(138,144),(139,144),(140,144),(141,144),(142,144),(143,144)],145)
=> ? = 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,18),(1,28),(1,30),(1,32),(1,35),(1,36),(1,72),(2,12),(2,17),(2,27),(2,29),(2,32),(2,33),(2,34),(2,71),(3,17),(3,20),(3,21),(3,30),(3,31),(3,37),(3,38),(3,73),(4,18),(4,19),(4,22),(4,29),(4,31),(4,39),(4,40),(4,74),(5,16),(5,24),(5,26),(5,34),(5,36),(5,38),(5,40),(5,75),(6,15),(6,23),(6,25),(6,33),(6,35),(6,37),(6,39),(6,75),(7,11),(7,15),(7,16),(7,19),(7,20),(7,71),(7,72),(8,14),(8,22),(8,23),(8,24),(8,27),(8,72),(8,73),(9,14),(9,21),(9,25),(9,26),(9,28),(9,71),(9,74),(10,11),(10,12),(10,13),(10,73),(10,74),(10,75),(11,84),(11,99),(11,100),(12,41),(12,82),(12,85),(12,99),(13,41),(13,83),(13,86),(13,100),(14,58),(14,59),(14,101),(14,102),(15,54),(15,56),(15,84),(15,87),(15,89),(16,55),(16,57),(16,84),(16,88),(16,90),(17,44),(17,45),(17,85),(17,91),(17,98),(18,46),(18,47),(18,86),(18,92),(18,98),(19,54),(19,55),(19,68),(19,92),(19,99),(20,56),(20,57),(20,68),(20,91),(20,100),(21,64),(21,65),(21,70),(21,91),(21,102),(22,66),(22,67),(22,69),(22,92),(22,102),(23,58),(23,60),(23,66),(23,89),(23,103),(24,59),(24,61),(24,67),(24,90),(24,103),(25,58),(25,62),(25,64),(25,87),(25,104),(26,59),(26,63),(26,65),(26,88),(26,104),(27,60),(27,61),(27,69),(27,85),(27,101),(28,62),(28,63),(28,70),(28,86),(28,101),(29,50),(29,51),(29,69),(29,98),(29,99),(30,52),(30,53),(30,70),(30,98),(30,100),(31,48),(31,49),(31,68),(31,98),(31,102),(32,41),(32,42),(32,43),(32,98),(32,101),(33,42),(33,44),(33,50),(33,60),(33,82),(33,87),(34,43),(34,45),(34,51),(34,61),(34,82),(34,88),(35,42),(35,46),(35,52),(35,62),(35,83),(35,89),(36,43),(36,47),(36,53),(36,63),(36,83),(36,90),(37,44),(37,48),(37,52),(37,56),(37,64),(37,103),(38,45),(38,49),(38,53),(38,57),(38,65),(38,103),(39,46),(39,48),(39,50),(39,54),(39,66),(39,104),(40,47),(40,49),(40,51),(40,55),(40,67),(40,104),(41,97),(41,114),(42,97),(42,105),(42,107),(43,97),(43,106),(43,108),(44,93),(44,105),(44,109),(45,94),(45,106),(45,109),(46,95),(46,105),(46,110),(47,96),(47,106),(47,110),(48,80),(48,105),(48,113),(49,81),(49,106),(49,113),(50,76),(50,105),(50,111),(51,77),(51,106),(51,111),(52,78),(52,105),(52,112),(53,79),(53,106),(53,112),(54,80),(54,95),(54,111),(55,81),(55,96),(55,111),(56,80),(56,93),(56,112),(57,81),(57,94),(57,112),(58,107),(58,113),(59,108),(59,113),(60,76),(60,107),(60,109),(61,77),(61,108),(61,109),(62,78),(62,107),(62,110),(63,79),(63,108),(63,110),(64,78),(64,93),(64,113),(65,79),(65,94),(65,113),(66,76),(66,95),(66,113),(67,77),(67,96),(67,113),(68,80),(68,81),(68,114),(69,76),(69,77),(69,114),(70,78),(70,79),(70,114),(71,87),(71,88),(71,91),(71,99),(71,101),(72,89),(72,90),(72,92),(72,100),(72,101),(73,85),(73,100),(73,102),(73,103),(74,86),(74,99),(74,102),(74,104),(75,82),(75,83),(75,84),(75,103),(75,104),(76,115),(77,115),(78,115),(79,115),(80,115),(81,115),(82,97),(82,109),(82,111),(83,97),(83,110),(83,112),(84,111),(84,112),(85,109),(85,114),(86,110),(86,114),(87,93),(87,107),(87,111),(88,94),(88,108),(88,111),(89,95),(89,107),(89,112),(90,96),(90,108),(90,112),(91,93),(91,94),(91,114),(92,95),(92,96),(92,114),(93,115),(94,115),(95,115),(96,115),(97,115),(98,105),(98,106),(98,114),(99,111),(99,114),(100,112),(100,114),(101,107),(101,108),(101,114),(102,113),(102,114),(103,109),(103,112),(103,113),(104,110),(104,111),(104,113),(105,115),(106,115),(107,115),(108,115),(109,115),(110,115),(111,115),(112,115),(113,115),(114,115)],116)
=> ? = 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,16),(1,17),(1,18),(1,34),(1,35),(1,40),(1,41),(1,42),(1,43),(2,14),(2,15),(2,18),(2,32),(2,33),(2,36),(2,37),(2,38),(2,39),(3,12),(3,21),(3,23),(3,27),(3,31),(3,37),(3,41),(3,79),(4,12),(4,20),(4,22),(4,26),(4,30),(4,36),(4,40),(4,78),(5,13),(5,20),(5,24),(5,28),(5,31),(5,38),(5,42),(5,76),(6,13),(6,21),(6,25),(6,29),(6,30),(6,39),(6,43),(6,77),(7,19),(7,28),(7,29),(7,33),(7,35),(7,78),(7,79),(8,19),(8,26),(8,27),(8,32),(8,34),(8,76),(8,77),(9,11),(9,14),(9,16),(9,22),(9,25),(9,76),(9,79),(10,11),(10,15),(10,17),(10,23),(10,24),(10,77),(10,78),(11,105),(11,106),(11,107),(12,44),(12,106),(12,108),(12,111),(13,45),(13,105),(13,108),(13,112),(14,60),(14,63),(14,89),(14,92),(14,107),(15,61),(15,62),(15,90),(15,91),(15,107),(16,64),(16,67),(16,93),(16,96),(16,107),(17,65),(17,66),(17,94),(17,95),(17,107),(18,46),(18,47),(18,107),(18,111),(18,112),(19,48),(19,49),(19,109),(19,110),(20,68),(20,70),(20,85),(20,87),(20,108),(21,69),(21,71),(21,86),(21,88),(21,108),(22,50),(22,60),(22,64),(22,85),(22,106),(23,51),(23,61),(23,65),(23,86),(23,106),(24,51),(24,62),(24,66),(24,87),(24,105),(25,50),(25,63),(25,67),(25,88),(25,105),(26,44),(26,52),(26,56),(26,85),(26,109),(27,44),(27,53),(27,57),(27,86),(27,110),(28,45),(28,54),(28,58),(28,87),(28,110),(29,45),(29,55),(29,59),(29,88),(29,109),(30,50),(30,72),(30,74),(30,108),(30,109),(31,51),(31,73),(31,75),(31,108),(31,110),(32,46),(32,48),(32,52),(32,53),(32,89),(32,90),(33,47),(33,48),(33,54),(33,55),(33,91),(33,92),(34,46),(34,49),(34,56),(34,57),(34,93),(34,94),(35,47),(35,49),(35,58),(35,59),(35,95),(35,96),(36,52),(36,60),(36,68),(36,72),(36,91),(36,111),(37,53),(37,61),(37,69),(37,73),(37,92),(37,111),(38,54),(38,62),(38,68),(38,73),(38,89),(38,112),(39,55),(39,63),(39,69),(39,72),(39,90),(39,112),(40,56),(40,64),(40,70),(40,74),(40,95),(40,111),(41,57),(41,65),(41,71),(41,75),(41,96),(41,111),(42,58),(42,66),(42,70),(42,75),(42,93),(42,112),(43,59),(43,67),(43,71),(43,74),(43,94),(43,112),(44,113),(44,122),(45,114),(45,122),(46,84),(46,113),(46,120),(47,84),(47,114),(47,121),(48,84),(48,115),(48,116),(49,84),(49,117),(49,118),(50,80),(50,82),(50,122),(51,81),(51,83),(51,122),(52,97),(52,113),(52,115),(53,98),(53,113),(53,116),(54,99),(54,114),(54,116),(55,100),(55,114),(55,115),(56,101),(56,113),(56,117),(57,102),(57,113),(57,118),(58,103),(58,114),(58,118),(59,104),(59,114),(59,117),(60,80),(60,97),(60,121),(61,81),(61,98),(61,121),(62,81),(62,99),(62,120),(63,80),(63,100),(63,120),(64,82),(64,101),(64,121),(65,83),(65,102),(65,121),(66,83),(66,103),(66,120),(67,82),(67,104),(67,120),(68,97),(68,99),(68,119),(69,98),(69,100),(69,119),(70,101),(70,103),(70,119),(71,102),(71,104),(71,119),(72,80),(72,115),(72,119),(73,81),(73,116),(73,119),(74,82),(74,117),(74,119),(75,83),(75,118),(75,119),(76,85),(76,89),(76,93),(76,105),(76,110),(77,86),(77,90),(77,94),(77,105),(77,109),(78,87),(78,91),(78,95),(78,106),(78,109),(79,88),(79,92),(79,96),(79,106),(79,110),(80,123),(81,123),(82,123),(83,123),(84,123),(85,97),(85,101),(85,122),(86,98),(86,102),(86,122),(87,99),(87,103),(87,122),(88,100),(88,104),(88,122),(89,97),(89,116),(89,120),(90,98),(90,115),(90,120),(91,99),(91,115),(91,121),(92,100),(92,116),(92,121),(93,101),(93,118),(93,120),(94,102),(94,117),(94,120),(95,103),(95,117),(95,121),(96,104),(96,118),(96,121),(97,123),(98,123),(99,123),(100,123),(101,123),(102,123),(103,123),(104,123),(105,120),(105,122),(106,121),(106,122),(107,120),(107,121),(108,119),(108,122),(109,115),(109,117),(109,122),(110,116),(110,118),(110,122),(111,113),(111,119),(111,121),(112,114),(112,119),(112,120),(113,123),(114,123),(115,123),(116,123),(117,123),(118,123),(119,123),(120,123),(121,123),(122,123)],124)
=> ? = 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,21),(1,25),(1,29),(1,33),(1,55),(1,57),(2,20),(2,24),(2,28),(2,32),(2,55),(2,56),(3,22),(3,26),(3,30),(3,32),(3,54),(3,57),(4,23),(4,27),(4,31),(4,33),(4,54),(4,56),(5,11),(5,14),(5,15),(5,17),(5,22),(5,23),(5,55),(6,11),(6,12),(6,13),(6,16),(6,20),(6,21),(6,54),(7,10),(7,12),(7,14),(7,18),(7,24),(7,27),(7,57),(8,10),(8,13),(8,15),(8,19),(8,25),(8,26),(8,56),(9,16),(9,17),(9,18),(9,19),(9,28),(9,29),(9,30),(9,31),(10,53),(10,60),(10,61),(10,74),(11,52),(11,58),(11,59),(11,74),(12,34),(12,38),(12,74),(12,76),(13,35),(13,39),(13,74),(13,75),(14,37),(14,40),(14,74),(14,77),(15,36),(15,41),(15,74),(15,78),(16,38),(16,39),(16,44),(16,45),(16,52),(16,62),(17,40),(17,41),(17,46),(17,47),(17,52),(17,63),(18,38),(18,40),(18,48),(18,51),(18,53),(18,65),(19,39),(19,41),(19,49),(19,50),(19,53),(19,64),(20,34),(20,44),(20,58),(20,75),(21,35),(21,45),(21,58),(21,76),(22,36),(22,46),(22,59),(22,77),(23,37),(23,47),(23,59),(23,78),(24,34),(24,48),(24,60),(24,77),(25,35),(25,49),(25,61),(25,78),(26,36),(26,50),(26,61),(26,75),(27,37),(27,51),(27,60),(27,76),(28,42),(28,44),(28,48),(28,63),(28,64),(29,43),(29,45),(29,49),(29,63),(29,65),(30,42),(30,46),(30,50),(30,62),(30,65),(31,43),(31,47),(31,51),(31,62),(31,64),(32,42),(32,75),(32,77),(33,43),(33,76),(33,78),(34,66),(34,84),(35,67),(35,84),(36,68),(36,84),(37,69),(37,84),(38,66),(38,80),(38,83),(39,67),(39,79),(39,83),(40,69),(40,81),(40,83),(41,68),(41,82),(41,83),(42,79),(42,81),(43,80),(43,82),(44,66),(44,70),(44,79),(45,67),(45,70),(45,80),(46,68),(46,71),(46,81),(47,69),(47,71),(47,82),(48,66),(48,72),(48,81),(49,67),(49,73),(49,82),(50,68),(50,73),(50,79),(51,69),(51,72),(51,80),(52,70),(52,71),(52,83),(53,72),(53,73),(53,83),(54,59),(54,62),(54,75),(54,76),(55,58),(55,63),(55,77),(55,78),(56,60),(56,64),(56,75),(56,78),(57,61),(57,65),(57,76),(57,77),(58,70),(58,84),(59,71),(59,84),(60,72),(60,84),(61,73),(61,84),(62,71),(62,79),(62,80),(63,70),(63,81),(63,82),(64,72),(64,79),(64,82),(65,73),(65,80),(65,81),(66,85),(67,85),(68,85),(69,85),(70,85),(71,85),(72,85),(73,85),(74,83),(74,84),(75,79),(75,84),(76,80),(76,84),(77,81),(77,84),(78,82),(78,84),(79,85),(80,85),(81,85),(82,85),(83,85),(84,85)],86)
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,21),(1,25),(1,29),(1,33),(1,55),(1,57),(2,20),(2,24),(2,28),(2,32),(2,55),(2,56),(3,22),(3,26),(3,30),(3,32),(3,54),(3,57),(4,23),(4,27),(4,31),(4,33),(4,54),(4,56),(5,11),(5,14),(5,15),(5,17),(5,22),(5,23),(5,55),(6,11),(6,12),(6,13),(6,16),(6,20),(6,21),(6,54),(7,10),(7,12),(7,14),(7,18),(7,24),(7,27),(7,57),(8,10),(8,13),(8,15),(8,19),(8,25),(8,26),(8,56),(9,16),(9,17),(9,18),(9,19),(9,28),(9,29),(9,30),(9,31),(10,53),(10,60),(10,61),(10,74),(11,52),(11,58),(11,59),(11,74),(12,34),(12,38),(12,74),(12,76),(13,35),(13,39),(13,74),(13,75),(14,37),(14,40),(14,74),(14,77),(15,36),(15,41),(15,74),(15,78),(16,38),(16,39),(16,44),(16,45),(16,52),(16,62),(17,40),(17,41),(17,46),(17,47),(17,52),(17,63),(18,38),(18,40),(18,48),(18,51),(18,53),(18,65),(19,39),(19,41),(19,49),(19,50),(19,53),(19,64),(20,34),(20,44),(20,58),(20,75),(21,35),(21,45),(21,58),(21,76),(22,36),(22,46),(22,59),(22,77),(23,37),(23,47),(23,59),(23,78),(24,34),(24,48),(24,60),(24,77),(25,35),(25,49),(25,61),(25,78),(26,36),(26,50),(26,61),(26,75),(27,37),(27,51),(27,60),(27,76),(28,42),(28,44),(28,48),(28,63),(28,64),(29,43),(29,45),(29,49),(29,63),(29,65),(30,42),(30,46),(30,50),(30,62),(30,65),(31,43),(31,47),(31,51),(31,62),(31,64),(32,42),(32,75),(32,77),(33,43),(33,76),(33,78),(34,66),(34,84),(35,67),(35,84),(36,68),(36,84),(37,69),(37,84),(38,66),(38,80),(38,83),(39,67),(39,79),(39,83),(40,69),(40,81),(40,83),(41,68),(41,82),(41,83),(42,79),(42,81),(43,80),(43,82),(44,66),(44,70),(44,79),(45,67),(45,70),(45,80),(46,68),(46,71),(46,81),(47,69),(47,71),(47,82),(48,66),(48,72),(48,81),(49,67),(49,73),(49,82),(50,68),(50,73),(50,79),(51,69),(51,72),(51,80),(52,70),(52,71),(52,83),(53,72),(53,73),(53,83),(54,59),(54,62),(54,75),(54,76),(55,58),(55,63),(55,77),(55,78),(56,60),(56,64),(56,75),(56,78),(57,61),(57,65),(57,76),(57,77),(58,70),(58,84),(59,71),(59,84),(60,72),(60,84),(61,73),(61,84),(62,71),(62,79),(62,80),(63,70),(63,81),(63,82),(64,72),(64,79),(64,82),(65,73),(65,80),(65,81),(66,85),(67,85),(68,85),(69,85),(70,85),(71,85),(72,85),(73,85),(74,83),(74,84),(75,79),(75,84),(76,80),(76,84),(77,81),(77,84),(78,82),(78,84),(79,85),(80,85),(81,85),(82,85),(83,85),(84,85)],86)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,15),(1,16),(1,17),(1,18),(1,54),(1,55),(2,10),(2,13),(2,14),(2,20),(2,55),(2,57),(3,10),(3,11),(3,12),(3,19),(3,54),(3,56),(4,12),(4,16),(4,23),(4,24),(4,26),(4,30),(4,57),(5,11),(5,15),(5,21),(5,22),(5,25),(5,29),(5,57),(6,14),(6,18),(6,22),(6,24),(6,28),(6,32),(6,56),(7,13),(7,17),(7,21),(7,23),(7,27),(7,31),(7,56),(8,19),(8,27),(8,28),(8,29),(8,30),(8,33),(8,55),(9,20),(9,25),(9,26),(9,31),(9,32),(9,33),(9,54),(10,58),(10,59),(10,78),(11,42),(11,58),(11,60),(11,64),(12,43),(12,58),(12,61),(12,65),(13,44),(13,59),(13,62),(13,66),(14,45),(14,59),(14,63),(14,67),(15,38),(15,39),(15,60),(15,80),(16,40),(16,41),(16,61),(16,80),(17,38),(17,40),(17,62),(17,79),(18,39),(18,41),(18,63),(18,79),(19,42),(19,43),(19,68),(19,78),(20,44),(20,45),(20,69),(20,78),(21,38),(21,46),(21,50),(21,64),(21,66),(22,39),(22,47),(22,51),(22,64),(22,67),(23,40),(23,48),(23,52),(23,65),(23,66),(24,41),(24,49),(24,53),(24,65),(24,67),(25,34),(25,50),(25,51),(25,60),(25,69),(26,35),(26,52),(26,53),(26,61),(26,69),(27,36),(27,46),(27,48),(27,62),(27,68),(28,37),(28,47),(28,49),(28,63),(28,68),(29,34),(29,42),(29,46),(29,47),(29,80),(30,35),(30,43),(30,48),(30,49),(30,80),(31,36),(31,44),(31,50),(31,52),(31,79),(32,37),(32,45),(32,51),(32,53),(32,79),(33,34),(33,35),(33,36),(33,37),(33,78),(34,70),(34,71),(34,86),(35,72),(35,73),(35,86),(36,70),(36,72),(36,87),(37,71),(37,73),(37,87),(38,82),(38,84),(39,82),(39,85),(40,83),(40,84),(41,83),(41,85),(42,74),(42,86),(43,75),(43,86),(44,76),(44,87),(45,77),(45,87),(46,70),(46,74),(46,84),(47,71),(47,74),(47,85),(48,72),(48,75),(48,84),(49,73),(49,75),(49,85),(50,70),(50,76),(50,82),(51,71),(51,77),(51,82),(52,72),(52,76),(52,83),(53,73),(53,77),(53,83),(54,60),(54,61),(54,78),(54,79),(55,62),(55,63),(55,78),(55,80),(56,59),(56,64),(56,65),(56,68),(56,79),(57,58),(57,66),(57,67),(57,69),(57,80),(58,81),(58,86),(59,81),(59,87),(60,82),(60,86),(61,83),(61,86),(62,84),(62,87),(63,85),(63,87),(64,74),(64,81),(64,82),(65,75),(65,81),(65,83),(66,76),(66,81),(66,84),(67,77),(67,81),(67,85),(68,74),(68,75),(68,87),(69,76),(69,77),(69,86),(70,88),(71,88),(72,88),(73,88),(74,88),(75,88),(76,88),(77,88),(78,86),(78,87),(79,82),(79,83),(79,87),(80,84),(80,85),(80,86),(81,88),(82,88),(83,88),(84,88),(85,88),(86,88),(87,88)],89)
=> ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,17),(1,18),(1,21),(1,22),(1,30),(1,32),(1,35),(1,36),(2,15),(2,16),(2,19),(2,20),(2,30),(2,31),(2,33),(2,34),(3,26),(3,27),(3,28),(3,29),(3,31),(3,32),(3,68),(4,12),(4,23),(4,25),(4,29),(4,34),(4,36),(4,70),(5,11),(5,23),(5,24),(5,28),(5,33),(5,35),(5,69),(6,10),(6,14),(6,19),(6,21),(6,24),(6,27),(6,70),(7,10),(7,13),(7,20),(7,22),(7,25),(7,26),(7,69),(8,11),(8,13),(8,15),(8,17),(8,68),(8,70),(9,12),(9,14),(9,16),(9,18),(9,68),(9,69),(10,39),(10,77),(10,78),(10,92),(11,62),(11,64),(11,79),(11,93),(12,63),(12,65),(12,80),(12,94),(13,58),(13,60),(13,78),(13,93),(14,59),(14,61),(14,77),(14,94),(15,58),(15,62),(15,66),(15,82),(15,85),(16,59),(16,63),(16,67),(16,81),(16,85),(17,60),(17,64),(17,66),(17,84),(17,86),(18,61),(18,65),(18,67),(18,83),(18,86),(19,49),(19,52),(19,59),(19,82),(19,92),(20,48),(20,53),(20,58),(20,81),(20,92),(21,51),(21,54),(21,61),(21,84),(21,92),(22,50),(22,55),(22,60),(22,83),(22,92),(23,37),(23,40),(23,41),(23,79),(23,80),(24,52),(24,54),(24,56),(24,77),(24,79),(25,53),(25,55),(25,57),(25,78),(25,80),(26,39),(26,48),(26,50),(26,57),(26,93),(27,39),(27,49),(27,51),(27,56),(27,94),(28,37),(28,44),(28,46),(28,56),(28,93),(29,37),(29,45),(29,47),(29,57),(29,94),(30,38),(30,42),(30,43),(30,66),(30,67),(30,92),(31,38),(31,44),(31,45),(31,48),(31,49),(31,85),(32,38),(32,46),(32,47),(32,50),(32,51),(32,86),(33,40),(33,42),(33,44),(33,52),(33,62),(33,81),(34,40),(34,43),(34,45),(34,53),(34,63),(34,82),(35,41),(35,42),(35,46),(35,54),(35,64),(35,83),(36,41),(36,43),(36,47),(36,55),(36,65),(36,84),(37,100),(37,103),(38,87),(38,95),(38,100),(39,95),(39,103),(40,88),(40,89),(40,100),(41,90),(41,91),(41,100),(42,75),(42,100),(42,101),(43,76),(43,100),(43,102),(44,71),(44,96),(44,100),(45,72),(45,97),(45,100),(46,73),(46,98),(46,100),(47,74),(47,99),(47,100),(48,72),(48,95),(48,96),(49,71),(49,95),(49,97),(50,74),(50,95),(50,98),(51,73),(51,95),(51,99),(52,71),(52,88),(52,101),(53,72),(53,89),(53,102),(54,73),(54,90),(54,101),(55,74),(55,91),(55,102),(56,71),(56,73),(56,103),(57,72),(57,74),(57,103),(58,96),(58,102),(59,97),(59,101),(60,98),(60,102),(61,99),(61,101),(62,75),(62,88),(62,96),(63,76),(63,89),(63,97),(64,75),(64,90),(64,98),(65,76),(65,91),(65,99),(66,75),(66,87),(66,102),(67,76),(67,87),(67,101),(68,85),(68,86),(68,93),(68,94),(69,77),(69,80),(69,81),(69,83),(69,93),(70,78),(70,79),(70,82),(70,84),(70,94),(71,104),(72,104),(73,104),(74,104),(75,104),(76,104),(77,101),(77,103),(78,102),(78,103),(79,88),(79,90),(79,103),(80,89),(80,91),(80,103),(81,89),(81,96),(81,101),(82,88),(82,97),(82,102),(83,91),(83,98),(83,101),(84,90),(84,99),(84,102),(85,87),(85,96),(85,97),(86,87),(86,98),(86,99),(87,104),(88,104),(89,104),(90,104),(91,104),(92,95),(92,101),(92,102),(93,96),(93,98),(93,103),(94,97),(94,99),(94,103),(95,104),(96,104),(97,104),(98,104),(99,104),(100,104),(101,104),(102,104),(103,104)],105)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,11),(2,15),(2,20),(2,21),(2,23),(2,50),(3,10),(3,14),(3,18),(3,19),(3,22),(3,50),(4,13),(4,17),(4,19),(4,21),(4,25),(4,49),(5,12),(5,16),(5,18),(5,20),(5,24),(5,49),(6,14),(6,15),(6,16),(6,17),(6,27),(6,48),(7,10),(7,11),(7,12),(7,13),(7,26),(7,48),(8,9),(8,48),(8,49),(8,50),(9,59),(9,60),(9,61),(10,28),(10,40),(10,41),(10,63),(11,29),(11,42),(11,43),(11,63),(12,30),(12,40),(12,42),(12,64),(13,31),(13,41),(13,43),(13,64),(14,32),(14,44),(14,45),(14,63),(15,33),(15,46),(15,47),(15,63),(16,34),(16,44),(16,46),(16,64),(17,35),(17,45),(17,47),(17,64),(18,36),(18,40),(18,44),(18,62),(19,37),(19,41),(19,45),(19,62),(20,38),(20,42),(20,46),(20,62),(21,39),(21,43),(21,47),(21,62),(22,28),(22,32),(22,36),(22,37),(22,59),(23,29),(23,33),(23,38),(23,39),(23,59),(24,30),(24,34),(24,36),(24,38),(24,60),(25,31),(25,35),(25,37),(25,39),(25,60),(26,28),(26,29),(26,30),(26,31),(26,61),(27,32),(27,33),(27,34),(27,35),(27,61),(28,51),(28,52),(28,66),(29,53),(29,54),(29,66),(30,51),(30,53),(30,67),(31,52),(31,54),(31,67),(32,55),(32,56),(32,66),(33,57),(33,58),(33,66),(34,55),(34,57),(34,67),(35,56),(35,58),(35,67),(36,51),(36,55),(36,65),(37,52),(37,56),(37,65),(38,53),(38,57),(38,65),(39,54),(39,58),(39,65),(40,51),(40,68),(41,52),(41,68),(42,53),(42,68),(43,54),(43,68),(44,55),(44,68),(45,56),(45,68),(46,57),(46,68),(47,58),(47,68),(48,61),(48,63),(48,64),(49,60),(49,62),(49,64),(50,59),(50,62),(50,63),(51,69),(52,69),(53,69),(54,69),(55,69),(56,69),(57,69),(58,69),(59,65),(59,66),(60,65),(60,67),(61,66),(61,67),(62,65),(62,68),(63,66),(63,68),(64,67),(64,68),(65,69),(66,69),(67,69),(68,69)],70)
=> ? = 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(1,13),(1,14),(1,44),(1,46),(2,9),(2,11),(2,12),(2,44),(2,45),(3,15),(3,24),(3,25),(3,26),(3,27),(3,44),(4,12),(4,17),(4,22),(4,23),(4,25),(4,46),(5,11),(5,16),(5,20),(5,21),(5,24),(5,46),(6,14),(6,19),(6,21),(6,23),(6,27),(6,45),(7,13),(7,18),(7,20),(7,22),(7,26),(7,45),(8,9),(8,10),(8,15),(8,16),(8,17),(8,18),(8,19),(9,36),(9,37),(9,47),(9,52),(10,38),(10,39),(10,47),(10,53),(11,36),(11,48),(11,62),(12,37),(12,49),(12,62),(13,38),(13,50),(13,63),(14,39),(14,51),(14,63),(15,32),(15,33),(15,34),(15,35),(15,47),(16,32),(16,36),(16,40),(16,41),(16,53),(17,33),(17,37),(17,42),(17,43),(17,53),(18,34),(18,38),(18,40),(18,42),(18,52),(19,35),(19,39),(19,41),(19,43),(19,52),(20,28),(20,40),(20,48),(20,50),(21,29),(21,41),(21,48),(21,51),(22,30),(22,42),(22,49),(22,50),(23,31),(23,43),(23,49),(23,51),(24,28),(24,29),(24,32),(24,62),(25,30),(25,31),(25,33),(25,62),(26,28),(26,30),(26,34),(26,63),(27,29),(27,31),(27,35),(27,63),(28,54),(28,66),(29,55),(29,66),(30,56),(30,66),(31,57),(31,66),(32,54),(32,55),(32,64),(33,56),(33,57),(33,64),(34,54),(34,56),(34,65),(35,55),(35,57),(35,65),(36,58),(36,64),(37,59),(37,64),(38,60),(38,65),(39,61),(39,65),(40,54),(40,58),(40,60),(41,55),(41,58),(41,61),(42,56),(42,59),(42,60),(43,57),(43,59),(43,61),(44,47),(44,62),(44,63),(45,48),(45,49),(45,52),(45,63),(46,50),(46,51),(46,53),(46,62),(47,64),(47,65),(48,58),(48,66),(49,59),(49,66),(50,60),(50,66),(51,61),(51,66),(52,58),(52,59),(52,65),(53,60),(53,61),(53,64),(54,67),(55,67),(56,67),(57,67),(58,67),(59,67),(60,67),(61,67),(62,64),(62,66),(63,65),(63,66),(64,67),(65,67),(66,67)],68)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,14),(1,19),(1,20),(1,21),(1,22),(1,35),(1,36),(2,10),(2,14),(2,15),(2,16),(2,17),(2,18),(2,33),(2,34),(3,13),(3,16),(3,20),(3,24),(3,26),(3,30),(3,70),(4,13),(4,15),(4,19),(4,23),(4,25),(4,29),(4,69),(5,12),(5,17),(5,21),(5,23),(5,27),(5,31),(5,70),(6,12),(6,18),(6,22),(6,24),(6,28),(6,32),(6,69),(7,29),(7,30),(7,31),(7,32),(7,34),(7,36),(7,68),(8,25),(8,26),(8,27),(8,28),(8,33),(8,35),(8,68),(9,10),(9,11),(9,68),(9,69),(9,70),(10,37),(10,79),(10,80),(10,83),(11,37),(11,81),(11,82),(11,84),(12,61),(12,63),(12,87),(12,88),(13,60),(13,62),(13,86),(13,88),(14,37),(14,38),(14,39),(14,86),(14,87),(15,40),(15,44),(15,56),(15,79),(15,86),(16,41),(16,45),(16,57),(16,80),(16,86),(17,42),(17,47),(17,56),(17,80),(17,87),(18,43),(18,46),(18,57),(18,79),(18,87),(19,48),(19,52),(19,58),(19,81),(19,86),(20,49),(20,53),(20,59),(20,82),(20,86),(21,50),(21,55),(21,58),(21,82),(21,87),(22,51),(22,54),(22,59),(22,81),(22,87),(23,56),(23,58),(23,64),(23,66),(23,88),(24,57),(24,59),(24,65),(24,67),(24,88),(25,40),(25,48),(25,60),(25,64),(25,89),(26,41),(26,49),(26,60),(26,65),(26,90),(27,42),(27,50),(27,61),(27,64),(27,90),(28,43),(28,51),(28,61),(28,65),(28,89),(29,44),(29,52),(29,62),(29,66),(29,89),(30,45),(30,53),(30,62),(30,67),(30,90),(31,47),(31,55),(31,63),(31,66),(31,90),(32,46),(32,54),(32,63),(32,67),(32,89),(33,38),(33,40),(33,41),(33,42),(33,43),(33,83),(34,39),(34,44),(34,45),(34,46),(34,47),(34,83),(35,38),(35,48),(35,49),(35,50),(35,51),(35,84),(36,39),(36,52),(36,53),(36,54),(36,55),(36,84),(37,85),(37,100),(38,85),(38,91),(38,92),(39,85),(39,93),(39,94),(40,71),(40,91),(40,95),(41,72),(41,91),(41,96),(42,71),(42,92),(42,96),(43,72),(43,92),(43,95),(44,73),(44,93),(44,95),(45,74),(45,93),(45,96),(46,74),(46,94),(46,95),(47,73),(47,94),(47,96),(48,75),(48,91),(48,97),(49,76),(49,91),(49,98),(50,75),(50,92),(50,98),(51,76),(51,92),(51,97),(52,77),(52,93),(52,97),(53,78),(53,93),(53,98),(54,78),(54,94),(54,97),(55,77),(55,94),(55,98),(56,71),(56,73),(56,100),(57,72),(57,74),(57,100),(58,75),(58,77),(58,100),(59,76),(59,78),(59,100),(60,91),(60,99),(61,92),(61,99),(62,93),(62,99),(63,94),(63,99),(64,71),(64,75),(64,99),(65,72),(65,76),(65,99),(66,73),(66,77),(66,99),(67,74),(67,78),(67,99),(68,83),(68,84),(68,89),(68,90),(69,79),(69,81),(69,88),(69,89),(70,80),(70,82),(70,88),(70,90),(71,101),(72,101),(73,101),(74,101),(75,101),(76,101),(77,101),(78,101),(79,95),(79,100),(80,96),(80,100),(81,97),(81,100),(82,98),(82,100),(83,85),(83,95),(83,96),(84,85),(84,97),(84,98),(85,101),(86,91),(86,93),(86,100),(87,92),(87,94),(87,100),(88,99),(88,100),(89,95),(89,97),(89,99),(90,96),(90,98),(90,99),(91,101),(92,101),(93,101),(94,101),(95,101),(96,101),(97,101),(98,101),(99,101),(100,101)],102)
=> ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(1,11),(1,12),(1,51),(1,52),(2,16),(2,20),(2,24),(2,28),(2,30),(2,52),(3,15),(3,19),(3,23),(3,28),(3,29),(3,51),(4,17),(4,21),(4,25),(4,27),(4,29),(4,52),(5,18),(5,22),(5,26),(5,27),(5,30),(5,51),(6,9),(6,11),(6,14),(6,19),(6,20),(6,21),(6,22),(7,9),(7,10),(7,13),(7,15),(7,16),(7,17),(7,18),(8,12),(8,13),(8,14),(8,23),(8,24),(8,25),(8,26),(9,37),(9,40),(9,69),(9,70),(10,37),(10,49),(10,53),(10,54),(11,37),(11,50),(11,55),(11,56),(12,49),(12,50),(12,57),(12,58),(13,40),(13,41),(13,42),(13,43),(13,44),(13,49),(14,40),(14,45),(14,46),(14,47),(14,48),(14,50),(15,33),(15,41),(15,53),(15,69),(16,34),(16,42),(16,54),(16,69),(17,33),(17,43),(17,54),(17,70),(18,34),(18,44),(18,53),(18,70),(19,35),(19,45),(19,55),(19,69),(20,36),(20,46),(20,56),(20,69),(21,35),(21,47),(21,56),(21,70),(22,36),(22,48),(22,55),(22,70),(23,31),(23,38),(23,41),(23,45),(23,57),(24,32),(24,38),(24,42),(24,46),(24,58),(25,31),(25,39),(25,43),(25,47),(25,58),(26,32),(26,39),(26,44),(26,48),(26,57),(27,39),(27,68),(27,70),(28,38),(28,68),(28,69),(29,31),(29,33),(29,35),(29,68),(30,32),(30,34),(30,36),(30,68),(31,60),(31,62),(31,71),(32,61),(32,63),(32,71),(33,60),(33,74),(34,61),(34,74),(35,62),(35,74),(36,63),(36,74),(37,59),(37,74),(38,71),(38,72),(39,71),(39,73),(40,59),(40,72),(40,73),(41,60),(41,64),(41,72),(42,61),(42,65),(42,72),(43,60),(43,65),(43,73),(44,61),(44,64),(44,73),(45,62),(45,66),(45,72),(46,63),(46,67),(46,72),(47,62),(47,67),(47,73),(48,63),(48,66),(48,73),(49,59),(49,64),(49,65),(50,59),(50,66),(50,67),(51,53),(51,55),(51,57),(51,68),(52,54),(52,56),(52,58),(52,68),(53,64),(53,74),(54,65),(54,74),(55,66),(55,74),(56,67),(56,74),(57,64),(57,66),(57,71),(58,65),(58,67),(58,71),(59,75),(60,75),(61,75),(62,75),(63,75),(64,75),(65,75),(66,75),(67,75),(68,71),(68,74),(69,72),(69,74),(70,73),(70,74),(71,75),(72,75),(73,75),(74,75)],76)
=> ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(1,11),(1,12),(1,51),(1,52),(2,16),(2,20),(2,24),(2,28),(2,30),(2,52),(3,15),(3,19),(3,23),(3,28),(3,29),(3,51),(4,17),(4,21),(4,25),(4,27),(4,29),(4,52),(5,18),(5,22),(5,26),(5,27),(5,30),(5,51),(6,9),(6,11),(6,14),(6,19),(6,20),(6,21),(6,22),(7,9),(7,10),(7,13),(7,15),(7,16),(7,17),(7,18),(8,12),(8,13),(8,14),(8,23),(8,24),(8,25),(8,26),(9,37),(9,40),(9,69),(9,70),(10,37),(10,49),(10,53),(10,54),(11,37),(11,50),(11,55),(11,56),(12,49),(12,50),(12,57),(12,58),(13,40),(13,41),(13,42),(13,43),(13,44),(13,49),(14,40),(14,45),(14,46),(14,47),(14,48),(14,50),(15,33),(15,41),(15,53),(15,69),(16,34),(16,42),(16,54),(16,69),(17,33),(17,43),(17,54),(17,70),(18,34),(18,44),(18,53),(18,70),(19,35),(19,45),(19,55),(19,69),(20,36),(20,46),(20,56),(20,69),(21,35),(21,47),(21,56),(21,70),(22,36),(22,48),(22,55),(22,70),(23,31),(23,38),(23,41),(23,45),(23,57),(24,32),(24,38),(24,42),(24,46),(24,58),(25,31),(25,39),(25,43),(25,47),(25,58),(26,32),(26,39),(26,44),(26,48),(26,57),(27,39),(27,68),(27,70),(28,38),(28,68),(28,69),(29,31),(29,33),(29,35),(29,68),(30,32),(30,34),(30,36),(30,68),(31,60),(31,62),(31,71),(32,61),(32,63),(32,71),(33,60),(33,74),(34,61),(34,74),(35,62),(35,74),(36,63),(36,74),(37,59),(37,74),(38,71),(38,72),(39,71),(39,73),(40,59),(40,72),(40,73),(41,60),(41,64),(41,72),(42,61),(42,65),(42,72),(43,60),(43,65),(43,73),(44,61),(44,64),(44,73),(45,62),(45,66),(45,72),(46,63),(46,67),(46,72),(47,62),(47,67),(47,73),(48,63),(48,66),(48,73),(49,59),(49,64),(49,65),(50,59),(50,66),(50,67),(51,53),(51,55),(51,57),(51,68),(52,54),(52,56),(52,58),(52,68),(53,64),(53,74),(54,65),(54,74),(55,66),(55,74),(56,67),(56,74),(57,64),(57,66),(57,71),(58,65),(58,67),(58,71),(59,75),(60,75),(61,75),(62,75),(63,75),(64,75),(65,75),(66,75),(67,75),(68,71),(68,74),(69,72),(69,74),(70,73),(70,74),(71,75),(72,75),(73,75),(74,75)],76)
=> ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,11),(2,15),(2,20),(2,21),(2,23),(2,50),(3,10),(3,14),(3,18),(3,19),(3,22),(3,50),(4,13),(4,17),(4,19),(4,21),(4,25),(4,49),(5,12),(5,16),(5,18),(5,20),(5,24),(5,49),(6,14),(6,15),(6,16),(6,17),(6,27),(6,48),(7,10),(7,11),(7,12),(7,13),(7,26),(7,48),(8,9),(8,48),(8,49),(8,50),(9,59),(9,60),(9,61),(10,28),(10,40),(10,41),(10,63),(11,29),(11,42),(11,43),(11,63),(12,30),(12,40),(12,42),(12,64),(13,31),(13,41),(13,43),(13,64),(14,32),(14,44),(14,45),(14,63),(15,33),(15,46),(15,47),(15,63),(16,34),(16,44),(16,46),(16,64),(17,35),(17,45),(17,47),(17,64),(18,36),(18,40),(18,44),(18,62),(19,37),(19,41),(19,45),(19,62),(20,38),(20,42),(20,46),(20,62),(21,39),(21,43),(21,47),(21,62),(22,28),(22,32),(22,36),(22,37),(22,59),(23,29),(23,33),(23,38),(23,39),(23,59),(24,30),(24,34),(24,36),(24,38),(24,60),(25,31),(25,35),(25,37),(25,39),(25,60),(26,28),(26,29),(26,30),(26,31),(26,61),(27,32),(27,33),(27,34),(27,35),(27,61),(28,51),(28,52),(28,66),(29,53),(29,54),(29,66),(30,51),(30,53),(30,67),(31,52),(31,54),(31,67),(32,55),(32,56),(32,66),(33,57),(33,58),(33,66),(34,55),(34,57),(34,67),(35,56),(35,58),(35,67),(36,51),(36,55),(36,65),(37,52),(37,56),(37,65),(38,53),(38,57),(38,65),(39,54),(39,58),(39,65),(40,51),(40,68),(41,52),(41,68),(42,53),(42,68),(43,54),(43,68),(44,55),(44,68),(45,56),(45,68),(46,57),(46,68),(47,58),(47,68),(48,61),(48,63),(48,64),(49,60),(49,62),(49,64),(50,59),(50,62),(50,63),(51,69),(52,69),(53,69),(54,69),(55,69),(56,69),(57,69),(58,69),(59,65),(59,66),(60,65),(60,67),(61,66),(61,67),(62,65),(62,68),(63,66),(63,68),(64,67),(64,68),(65,69),(66,69),(67,69),(68,69)],70)
=> ? = 2
([(0,5),(1,5),(2,3),(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)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],1)
=> 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,2),(0,3),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ([],1)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001625The Möbius invariant of a lattice. St001330The hat guessing number of a graph. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension 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. 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 $eA$. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001890The maximum magnitude of the Möbius function of a poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition.
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