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Your data matches 50 different statistics following compositions of up to 3 maps.
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Matching statistic: St000171
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St001118
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
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: St001674
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3 = 2 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 3 + 1
Description
The number of vertices of the largest induced star graph in the graph.
Matching statistic: St000271
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,5,3,7,6,4] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,5,4,7,3,6] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,2,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,4,2,6,7,5] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[1,3,4,2,7,5,6] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[1,3,4,7,2,6,5] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,4,7,5,2,6] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,5,2,4,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,5,4,7,2,6] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,6,2,5,4,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,6,4,2,5,7] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,3,7,2,4,6,5] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,7,2,5,4,6] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,3,7,4,2,5,6] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,4,2,3,6,7,5] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[1,4,2,3,7,5,6] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,4,2,5,7,6,3] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,4,2,6,5,3,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,4,2,6,5,7,3] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,4,3,5,7,2,6] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,4,3,6,2,5,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,4,3,7,2,5,6] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,5,2,3,7,6,4] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,5,2,4,6,3,7] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,5,2,4,6,7,3] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,5,3,2,6,4,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2
[1,5,3,2,6,7,4] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,6,2,3,5,7,4] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,6,2,4,3,7,5] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3
[1,6,3,2,4,7,5] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 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: St001578
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0 = 2 - 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0 = 2 - 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 3 - 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0 = 2 - 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 3 - 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0 = 2 - 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 3 - 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 2
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 3 - 2
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 2
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 2
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 - 2
[4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 3 - 2
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 2
[5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,4,5,6,3,7] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 2 - 2
[1,2,5,3,7,6,4] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,5,4,7,3,6] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,6,3,4,5,7] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
[1,3,2,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 - 2
Description
The minimal number of edges to add or remove to make a graph a line graph.
Matching statistic: St001642
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(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)
=> ? = 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,5),(0,6),(1,3),(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
[4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(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)
=> ? = 3
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,4,5,6,3,7] => ([(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
[1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 4
[1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[1,2,5,3,7,6,4] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,5,4,7,3,6] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,6,3,4,5,7] => ([(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
[1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 4
[1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,3,2,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
Description
The Prague dimension of a graph.
This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product.
Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
Matching statistic: St000718
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 + 1
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 3 + 1
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 2 + 1
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3 + 1
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 3 + 1
[2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001330
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> 5
[2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2 - 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 2 - 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 2 - 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2 - 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2 - 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(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)
=> ? = 2 - 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(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),(5,6)],7)
=> ? = 2 - 1
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> 4 = 5 - 1
[2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
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: St001804
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Values
[2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 2
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 2
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[2,3,4,5,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 4
[2,3,5,1,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 3
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[2,4,1,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[2,4,5,3,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[2,5,1,3,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 3
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[2,5,4,1,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[3,1,4,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 3
[3,1,5,2,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[3,4,2,5,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[3,5,2,1,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[4,1,2,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 3
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[4,1,5,3,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 2
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[4,3,1,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[5,1,2,3,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 4
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[5,1,4,2,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 3
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[5,3,1,2,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 2
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,4,5,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 3
[1,2,4,6,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,5,3,6,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 2
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[1,2,6,3,4,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 3
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,3,2,6,4,5] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,3,4,2,6,5] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,3,5,6,2,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,3,6,5,4,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,4,2,3,6,5] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,4,5,2,6,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,4,6,2,3,5] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001520The number of strict 3-descents. St000744The length of the path to the largest entry in a standard Young tableau. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000422The energy of a graph, if it is integral. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001644The dimension of a graph. St001964The interval resolution global dimension of a poset.
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