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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St001644
Values
([],1)
=> ([],0)
=> ([],1)
=> 0
([],2)
=> ([],0)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],0)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],0)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
([],5)
=> ([],0)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
([],6)
=> ([],0)
=> ([],1)
=> 0
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000271
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],2)
=> 0
([],2)
=> ([],2)
=> ([],3)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 1
([],3)
=> ([],3)
=> ([],4)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],4)
=> ([],5)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],5)
=> ([],5)
=> ([],6)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3
([],6)
=> ([],6)
=> ([],7)
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> ([(5,6)],7)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(4,6),(5,6)],7)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(3,6),(4,6),(5,6)],7)
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(3,6),(4,5)],7)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(2,3),(4,6),(5,6)],7)
=> ? = 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
([],7)
=> ([],7)
=> ([],8)
=> ? = 0
([(5,6)],7)
=> ([(5,6)],7)
=> ([(6,7)],8)
=> ? = 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(4,7),(5,7),(6,7)],8)
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([(4,7),(5,6)],8)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(4,7),(5,6),(6,7)],8)
=> 2
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([(3,4),(5,7),(6,7)],8)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,7),(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 3
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(1,7),(2,6),(3,5),(4,5),(6,7)],8)
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,7),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 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)
=> ([(2,6),(2,7),(3,4),(3,5),(4,7),(5,6)],8)
=> ? = 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,2),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 3
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000454
Values
([],1)
=> ([],0)
=> ([],1)
=> 0
([],2)
=> ([],0)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],0)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],0)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
([],5)
=> ([],0)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3
([],6)
=> ([],0)
=> ([],1)
=> 0
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([],7)
=> ([],0)
=> ([],1)
=> 0
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001117
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],2)
=> 0
([],2)
=> ([],3)
=> 0
([(0,1)],2)
=> ([(1,2)],3)
=> 1
([],3)
=> ([],4)
=> 0
([(1,2)],3)
=> ([(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],5)
=> 0
([(2,3)],4)
=> ([(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([],5)
=> ([],6)
=> 0
([(3,4)],5)
=> ([(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3
([],6)
=> ([],7)
=> ? = 0
([(4,5)],6)
=> ([(5,6)],7)
=> ? = 1
([(3,5),(4,5)],6)
=> ([(4,6),(5,6)],7)
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 5
([(2,5),(3,4)],6)
=> ([(3,6),(4,5)],7)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(2,3),(4,6),(5,6)],7)
=> ? = 2
([(3,4),(3,5),(4,5)],6)
=> ([(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? = 2
([],7)
=> ([],8)
=> ? = 0
([(5,6)],7)
=> ([(6,7)],8)
=> ? = 1
([(4,6),(5,6)],7)
=> ([(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([(4,7),(5,7),(6,7)],8)
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
([(3,6),(4,5)],7)
=> ([(4,7),(5,6)],8)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(3,4),(5,7),(6,7)],8)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,7),(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 3
([(1,6),(2,5),(3,4)],7)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(6,7)],8)
=> ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,3),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(1,7),(2,6),(3,5),(4,5),(6,7)],8)
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,7),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(4,7),(5,6)],8)
=> ? = 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? = 3
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St000264
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> ? = 0
([],2)
=> ([],2)
=> [2] => ([],2)
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> ? = 1
([],3)
=> ([],3)
=> [3] => ([],3)
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],4)
=> [4] => ([],4)
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],5)
=> ([],5)
=> [5] => ([],5)
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],6)
=> ([],6)
=> [6] => ([],6)
=> ? = 0
([(4,5)],6)
=> ([(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 5
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(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)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(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)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
([],7)
=> ([],7)
=> [7] => ([],7)
=> ? = 0
([(5,6)],7)
=> ([(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 6
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> [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)
=> 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> [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)
=> 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [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)
=> 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => ([(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)
=> 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 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: St001060
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([],2)
=> ([],1)
=> ? = 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1
([],3)
=> ([],3)
=> ([],1)
=> ? = 0
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ? = 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],4)
=> ([],1)
=> ? = 0
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 3
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 3
([],5)
=> ([],5)
=> ([],1)
=> ? = 0
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 4
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],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)
=> 3
([],6)
=> ([],6)
=> ([],1)
=> ? = 0
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 5
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 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)
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2
([],7)
=> ([],7)
=> ([],1)
=> ? = 0
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],1)
=> ([],1)
=> ? = 6
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2)],3)
=> ? = 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001118
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 0
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 0
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],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)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 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)
=> 3
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 0
([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 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),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 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),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? = 2
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 0
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 1
([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
([(3,6),(4,5)],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),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 1
([(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),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(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),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 3
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? = 2
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001603
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 26%●distinct values known / distinct values provided: 14%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 26%●distinct values known / distinct values provided: 14%
Values
([],1)
=> [1] => [1] => [1]
=> ? = 0 - 2
([],2)
=> [2] => [1] => [1]
=> ? = 0 - 2
([(0,1)],2)
=> [1,1] => [2] => [2]
=> ? = 1 - 2
([],3)
=> [3] => [1] => [1]
=> ? = 0 - 2
([(1,2)],3)
=> [2,1] => [1,1] => [1,1]
=> ? = 1 - 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1] => [1,1]
=> ? = 2 - 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => [3]
=> 1 = 3 - 2
([],4)
=> [4] => [1] => [1]
=> ? = 0 - 2
([(2,3)],4)
=> [3,1] => [1,1] => [1,1]
=> ? = 1 - 2
([(1,3),(2,3)],4)
=> [3,1] => [1,1] => [1,1]
=> ? = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1] => [1,1]
=> ? = 3 - 2
([(0,3),(1,2)],4)
=> [2,2] => [2] => [2]
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2] => [2]
=> ? = 2 - 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2] => [2]
=> ? = 3 - 2
([],5)
=> [5] => [1] => [1]
=> ? = 0 - 2
([(3,4)],5)
=> [4,1] => [1,1] => [1,1]
=> ? = 1 - 2
([(2,4),(3,4)],5)
=> [4,1] => [1,1] => [1,1]
=> ? = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1] => [1,1]
=> ? = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1] => [1,1]
=> ? = 4 - 2
([(1,4),(2,3)],5)
=> [3,2] => [1,1] => [1,1]
=> ? = 1 - 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1] => [1,1]
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,1] => [2,1]
=> 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1] => [2,1]
=> 1 = 3 - 2
([],6)
=> [6] => [1] => [1]
=> ? = 0 - 2
([(4,5)],6)
=> [5,1] => [1,1] => [1,1]
=> ? = 1 - 2
([(3,5),(4,5)],6)
=> [5,1] => [1,1] => [1,1]
=> ? = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1] => [1,1]
=> ? = 3 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1] => [1,1]
=> ? = 4 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1] => [1,1]
=> ? = 5 - 2
([(2,5),(3,4)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [2] => [2]
=> ? = 2 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [2] => [2]
=> ? = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [2] => [2]
=> ? = 3 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [3,3] => [2] => [2]
=> ? = 2 - 2
([],7)
=> [7] => [1] => [1]
=> ? = 0 - 2
([(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 1 - 2
([(4,6),(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 4 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 5 - 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => [1,1] => [1,1]
=> ? = 6 - 2
([(3,6),(4,5)],7)
=> [5,2] => [1,1] => [1,1]
=> ? = 1 - 2
([(3,6),(4,5),(5,6)],7)
=> [5,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> [5,2] => [1,1] => [1,1]
=> ? = 2 - 2
([(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [5,2] => [1,1] => [1,1]
=> ? = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => [1,2] => [2,1]
=> 1 = 3 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2] => [1,1] => [1,1]
=> ? = 4 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 1 = 3 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,3,1] => [2,1] => [2,1]
=> 1 = 3 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [3,3,1] => [2,1] => [2,1]
=> 1 = 3 - 2
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001570
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0 - 3
([],2)
=> ([],2)
=> ([],1)
=> ? = 0 - 3
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 - 3
([],3)
=> ([],3)
=> ([],1)
=> ? = 0 - 3
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ? = 1 - 3
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([],4)
=> ([],4)
=> ([],1)
=> ? = 0 - 3
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 3 - 3
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1 - 3
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 3 - 3
([],5)
=> ([],5)
=> ([],1)
=> ? = 0 - 3
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 3 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 3
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 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 - 3
([],6)
=> ([],6)
=> ([],1)
=> ? = 0 - 3
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 1 - 3
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 3 - 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 4 - 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 5 - 3
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 3 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(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),(1,2),(1,4),(2,3)],5)
=> 0 = 3 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 2 - 3
([],7)
=> ([],7)
=> ?
=> ? = 0 - 3
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 1 - 3
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 2 - 3
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 4 - 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 5 - 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],1)
=> ([],1)
=> ? = 6 - 3
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 1 - 3
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 3 - 3
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001604
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> ? = 0 - 2
([],2)
=> ([],2)
=> ([],1)
=> [1]
=> ? = 0 - 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([],3)
=> ([],3)
=> ([],1)
=> [1]
=> ? = 0 - 2
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([],4)
=> ([],4)
=> ([],1)
=> [1]
=> ? = 0 - 2
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 3 - 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 3 - 2
([],5)
=> ([],5)
=> ([],1)
=> [1]
=> ? = 0 - 2
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 4 - 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 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)
=> [5]
=> 1 = 3 - 2
([],6)
=> ([],6)
=> ([],1)
=> [1]
=> ? = 0 - 2
([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 3 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 4 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 5 - 2
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> [4]
=> 1 = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [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)
=> ([(0,1),(0,2),(1,2)],3)
=> [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),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(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)
=> [4]
=> 1 = 3 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2 - 2
([],7)
=> ([],7)
=> ?
=> ?
=> ? = 0 - 2
([(5,6)],7)
=> ([(5,6)],7)
=> ?
=> ?
=> ? = 1 - 2
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 5 - 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 6 - 2
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ?
=> ?
=> ? = 1 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ?
=> ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 3 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ?
=> ? = 2 - 2
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000284The Plancherel distribution on integer partitions. St000379The number of Hamiltonian cycles in a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000997The even-odd crank of an integer partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
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