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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000264
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Values
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,4),(2,3)],5)
=> ([(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)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 3
([(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)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> 4
([(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)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(2,5),(3,4)],6)
=> ([(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)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,5),(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),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 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)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(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),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 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)
=> ([(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)
=> 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)
=> ([(1,4),(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)
=> 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)
=> ([(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)
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 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)
=> ([(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)
=> 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)
=> ([(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)
=> 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)
=> ([(1,5),(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)
=> 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 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)
=> ([(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)
=> 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)
=> ([(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)
=> 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)
=> ([(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)
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001672
Values
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(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),(1,2)],3)
=> 1 = 3 - 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),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(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)],2)
=> 2 = 4 - 2
([(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)],2)
=> 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 5 - 2
([(2,5),(3,4)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,5),(2,3),(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 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),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 1 = 3 - 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),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(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),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(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),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 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),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 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)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(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),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,4),(1,2),(1,3),(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 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)
=> ([(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),(5,6)],7)
=> ?
=> ? = 3 - 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)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 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)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 3 - 2
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 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)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 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)
=> ([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,2),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(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)
=> ([(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(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)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 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),(2,5),(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)
=> ?
=> ? = 3 - 2
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,6),(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
([(0,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 2
Description
The restrained domination number of a graph.
This is the minimal size of a set of vertices $D$ such that every vertex not in $D$ is adjacent to a vertex in $D$ and to a vertex not in $D$.
Matching statistic: St001957
Values
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(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),(1,2)],3)
=> 0 = 3 - 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),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(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),(1,2)],3)
=> 0 = 3 - 3
([(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),(1,2)],3)
=> 0 = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(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)],2)
=> 1 = 4 - 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)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 5 - 3
([(2,5),(3,4)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,5),(2,3),(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(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),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(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),(1,2)],3)
=> 0 = 3 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(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),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 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),(1,2)],3)
=> 0 = 3 - 3
([(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),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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),(1,2)],3)
=> 0 = 3 - 3
([(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),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 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)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
([(0,4),(1,2),(1,3),(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(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),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 3 - 3
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,2),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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)
=> ([(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(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)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(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),(2,5),(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)
=> ?
=> ? = 3 - 3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,6),(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(0,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
Description
The number of Hasse diagrams with a given underlying undirected graph.
In particular, this statistic vanishes if the graph contains a triangle.
This is the size of the preimage of [[Mp00074]].
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
([(1,4),(2,3)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
([(2,5),(3,4)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(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)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(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)
=> [3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
([(0,6),(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)
=> [3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000781
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 2
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 2
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 2
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 2
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1 = 3 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St000205
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001175
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5 - 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 3 - 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> []
=> ? = 4 - 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 4 - 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3 - 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4 - 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? = 3 - 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0 = 3 - 3
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 3 - 3
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
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