- FindStat

Your data matches 79 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000264

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,5,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[5,2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,2,5,4,6] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,2,5,6,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,2,6,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,6,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,3,5,2,6] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,6,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,3,6,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,4,6,2,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,6,3,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,4,6] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,5,3,2,6,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,5,3,6,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,5,6,3,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St000741
(load all 4 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 4 - 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 4 - 3
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 4 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 4 - 3
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 4 - 3
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 1 = 4 - 3
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 3
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 3
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 1 = 4 - 3
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 3
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 1 = 4 - 3
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,4,6,5,3] => ([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,5,4,3,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 1 = 4 - 3
[2,1,5,4,6,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 4 - 3
[2,1,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,6,4,3,5] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,6,4,5,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 4 - 3
[2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 1 = 4 - 3
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ? = 4 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[2,6,1,4,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,6,1,5,3,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,6,4,1,3,5] => ([(0,4),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[3,1,2,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3
[3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 - 3
[3,1,5,6,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 3
[3,1,6,4,2,5] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 3
[3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 3

Description

The Colin de Verdière graph invariant.

Matching statistic: St001271

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1 = 4 - 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1 = 4 - 3
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1 = 4 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 4 - 3
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => 1 = 4 - 3
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 1 = 4 - 3
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 4 - 3
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 4 - 3
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 3
[3,2,5,1,6,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[3,6,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[5,6,1,3,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[5,6,2,1,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[6,1,3,4,2,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[6,1,4,2,3,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[6,2,4,1,5,3] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 4 - 3
[1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,3,7,5,4,6] => ([(0,4),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 4 - 3
[1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,1),(0,2),(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 3
[1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 3
[1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 3
[1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,2,4,6,3,7,5] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,6),(1,3),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3

Description

The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St001630
(load all 2 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4 - 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4 - 2
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 5 - 2
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 2
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 5 - 2
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 2
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 5 - 2
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 2
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4 - 2
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 5 - 2
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 5 - 2
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 2
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 5 - 2
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 4 - 2
[2,1,4,6,5,3] => ([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,5,4,3,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,5,4,6,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,6,4,3,5] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,6,4,5,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,1,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,6,3,1,5,4] => ([(0,4),(0,5),(1,2),(1,3),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,6,4,3,1,5] => ([(0,5),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,1,2,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,1,5,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,1,5,6,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,6,1,5,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[3,6,2,5,4,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[4,1,3,2,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,2,1,4,3,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,2,1,4,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[5,6,2,1,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2

Description

The global dimension of the incidence algebra of the lattice over the rational numbers.

Matching statistic: St000422

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,2,5,1,6,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[3,6,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[6,1,3,4,2,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,1,4,2,3,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,2,4,1,5,3] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 2
[1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,3,7,5,4,6] => ([(0,4),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,6,3,7,5] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2

Description

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.

Matching statistic: St000718

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 4 - 2
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[3,2,5,1,6,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[3,6,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 2
[6,1,3,4,2,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,1,4,2,3,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,2,4,1,5,3] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 2
[1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 2
[1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,3,7,5,4,6] => ([(0,4),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 2
[1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,6,3,7,5] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 2
[1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 2

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: St000455
(load all 3 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,2,6,4,3,5] => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[1,3,5,6,2,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,3,6,5,2,4] => ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,4,2,6,5,3] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[1,4,3,5,6,2] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,3,6,2,5] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,6,3,2,5] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,4,6,3,5,2] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[1,5,3,2,6,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,3,2,5,4] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,6,3,5,2,4] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,6,4,2,5,3] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[1,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => -1 = 4 - 5
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[3,1,6,2,5,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[3,2,5,1,6,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[3,6,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 5
[6,1,3,4,2,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[6,1,4,2,3,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[6,2,4,1,5,3] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 5 - 5
[1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 5
[1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 5
[1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 5
[1,2,3,7,5,4,6] => ([(0,4),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 5
[1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => ?
 => ? = 5 - 5
[1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 5
[1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,6,3,7,5] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 5
[1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
 => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 4 - 5

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001330
(load all 4 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => [5,4,2,3,1] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,3,2,5,4] => [4,5,2,3,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 4 - 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,5,3,2,4] => [4,2,3,5,1] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[2,1,4,3,5] => [5,3,4,1,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 4 - 2
[2,1,4,5,3] => [3,5,4,1,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[2,1,5,4,3] => [3,4,5,1,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 4 - 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[5,1,3,2,4] => [4,2,3,1,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 4 - 2
[5,2,1,4,3] => [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 4 - 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [4,2,1,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ? = 4 - 2
[1,2,4,3,6,5] => [5,6,3,4,2,1] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 2
[1,2,4,6,3,5] => [5,3,6,4,2,1] => [6,4,2,1,5,3] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,2,6,4,3,5] => [5,3,4,6,2,1] => [6,2,1,4,5,3] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,2,4,6,5] => [5,6,4,2,3,1] => [3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 4 - 2
[1,3,2,5,6,4] => [4,6,5,2,3,1] => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,2,6,4,5] => [5,4,6,2,3,1] => [3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 2
[1,3,2,6,5,4] => [4,5,6,2,3,1] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,3,4,2,6,5] => [5,6,2,4,3,1] => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,3,4,6,2,5] => [5,2,6,4,3,1] => [6,4,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,5,2,6,4] => [4,6,2,5,3,1] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,5,6,2,4] => [4,2,6,5,3,1] => [6,5,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [6,3,1,5,4,2] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,6,2,5,4] => [4,5,2,6,3,1] => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [6,3,1,4,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,3,6,5,2,4] => [4,2,5,6,3,1] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [4,1,6,5,3,2] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,4,2,3,6,5] => [5,6,3,2,4,1] => [4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 2
[1,4,2,5,3,6] => [6,3,5,2,4,1] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,2,5,6,3] => [3,6,5,2,4,1] => [4,1,6,5,2,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,2,6,3,5] => [5,3,6,2,4,1] => [4,1,6,2,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 5 - 2
[1,4,2,6,5,3] => [3,5,6,2,4,1] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 2
[1,4,3,2,5,6] => [6,5,2,3,4,1] => [4,1,3,6,5,2] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,3,2,6,5] => [5,6,2,3,4,1] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,3,5,2,6] => [6,2,5,3,4,1] => [4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,3,5,6,2] => [2,6,5,3,4,1] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,3,6,2,5] => [5,2,6,3,4,1] => [4,1,6,3,5,2] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,3,6,5,2] => [2,5,6,3,4,1] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,4,6,2,3,5] => [5,3,2,6,4,1] => [6,4,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,4,6,2,5,3] => [3,5,2,6,4,1] => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,6,3,2,5] => [5,2,3,6,4,1] => [6,4,1,3,5,2] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,4,6,3,5,2] => [2,5,3,6,4,1] => [6,4,1,2,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,5,3,2,4,6] => [6,4,2,3,5,1] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,5,3,2,6,4] => [4,6,2,3,5,1] => [5,1,3,6,2,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,5,3,6,2,4] => [4,2,6,3,5,1] => [5,1,6,3,4,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,5,6,3,2,4] => [4,2,3,6,5,1] => [6,5,1,3,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,2,4,3,5] => [5,3,4,2,6,1] => [6,1,4,2,5,3] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,3,2,4,5] => [5,4,2,3,6,1] => [6,1,3,5,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,3,2,5,4] => [4,5,2,3,6,1] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,3,4,2,5] => [5,2,4,3,6,1] => [6,1,4,3,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,6,3,5,2,4] => [4,2,5,3,6,1] => [6,1,5,3,4,2] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,4,2,3,5] => [5,3,2,4,6,1] => [6,1,4,5,3,2] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,6,4,2,5,3] => [3,5,2,4,6,1] => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,6,4,3,2,5] => [5,2,3,4,6,1] => [6,1,3,4,5,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[2,1,4,3,6,5] => [5,6,3,4,1,2] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 4 - 2
[2,1,5,4,6,3] => [3,6,4,5,1,2] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 4 - 2
[2,1,6,5,4,3] => [3,4,5,6,1,2] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[2,4,3,6,5,1] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 4 - 2
[3,2,1,5,4,6] => [6,4,5,1,2,3] => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 4 - 2
[3,2,1,6,5,4] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[3,2,5,4,6,1] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 4 - 2
[3,2,6,5,4,1] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[4,1,3,2,6,5] => [5,6,2,3,1,4] => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 4 - 2
[5,2,1,4,3,6] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 4 - 2
[5,6,1,3,2,4] => [4,2,3,1,6,5] => [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[5,6,2,1,4,3] => [3,4,1,2,6,5] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,1,3,2,5,4] => [4,5,2,3,1,6] => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 4 - 2
[6,1,4,3,5,2] => [2,5,3,4,1,6] => [4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,2,1,4,3,5] => [5,3,4,1,2,6] => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 4 - 2
[6,2,1,5,4,3] => [3,4,5,1,2,6] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,2,4,3,5,1] => [1,5,3,4,2,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,3,2,1,5,4] => [4,5,1,2,3,6] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,3,2,5,4,1] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,4,1,3,2,5] => [5,2,3,1,4,6] => [3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,5,1,3,2,4] => [4,2,3,1,5,6] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[6,5,2,1,4,3] => [3,4,1,2,5,6] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 4 - 2
[1,3,2,6,5,7,4] => [4,7,5,6,2,3,1] => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 4 - 2
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 4 - 2
[1,4,3,6,5,7,2] => [2,7,5,6,3,4,1] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,4,3,7,6,5,2] => [2,5,6,7,3,4,1] => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,5,4,7,6,3,2] => [2,3,6,7,4,5,1] => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 4 - 2
[2,1,4,3,6,5,7] => [7,5,6,3,4,1,2] => [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 4 - 2
[2,1,4,3,7,6,5] => [5,6,7,3,4,1,2] => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[2,1,5,4,7,6,3] => [3,6,7,4,5,1,2] => [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 4 - 2
[2,1,6,5,7,4,3] => [3,4,7,5,6,1,2] => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 4 - 2
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 4 - 2
[3,2,1,5,4,7,6] => [6,7,4,5,1,2,3] => [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[3,2,1,6,5,7,4] => [4,7,5,6,1,2,3] => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 4 - 2
[3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 4 - 2
[4,1,3,2,6,5,7] => [7,5,6,2,3,1,4] => [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 4 - 2
[4,1,3,2,7,6,5] => [5,6,7,2,3,1,4] => [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 4 - 2
[4,3,2,1,6,5,7] => [7,5,6,1,2,3,4] => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 4 - 2
[4,3,2,1,7,6,5] => [5,6,7,1,2,3,4] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 4 - 2
[5,1,4,3,7,6,2] => [2,6,7,3,4,1,5] => [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 4 - 2
[5,2,1,4,3,7,6] => [6,7,3,4,1,2,5] => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 4 - 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: St000068
(load all 2 compositions to match this statistic)

Mapping

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%

Values

[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 4 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 4 - 3
[1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 4 - 3
[1,5,3,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 4 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 4 - 3
[2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 4 - 3
[2,1,5,4,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 4 - 3
[2,5,1,4,3] => [5,2,4,1,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 4 - 3
[5,1,3,2,4] => [5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 4 - 3
[5,2,1,4,3] => [2,5,4,1,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 4 - 3
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,6,4,3,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 1 = 4 - 3
[1,2,6,4,3,5] => [1,2,4,6,3,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 5 - 3
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 5 - 3
[1,3,4,6,2,5] => [1,6,3,4,2,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 5 - 3
[1,3,5,2,4,6] => [1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 4 - 3
[1,3,5,2,6,4] => [1,5,3,6,2,4] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
 => ? = 4 - 3
[1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,2,6,4,3,5] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
 => ? = 4 - 3
[1,3,6,2,4,5] => [1,6,3,2,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 1 = 4 - 3
[1,3,6,2,5,4] => [1,6,3,5,2,4] => [1,2,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
 => ? = 4 - 3
[1,3,6,4,2,5] => [1,4,3,6,2,5] => [1,2,4,6,5,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 5 - 3
[1,3,6,5,2,4] => [1,5,3,2,6,4] => [1,2,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 4 - 3
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 5 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 5 - 3
[1,4,2,5,3,6] => [1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 4 - 3
[1,4,2,5,6,3] => [1,4,6,2,5,3] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
 => ? = 4 - 3
[1,4,2,6,3,5] => [1,4,6,2,3,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
 => ? = 5 - 3
[1,4,2,6,5,3] => [1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 4 - 3
[1,4,3,2,5,6] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,4,3,2,6,5] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,4,3,5,2,6] => [1,5,4,3,2,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
 => ? = 4 - 3
[1,4,3,5,6,2] => [1,6,4,3,5,2] => [1,2,6,3,4,5] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 4 - 3
[1,4,3,6,2,5] => [1,6,4,3,2,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 4 - 3
[1,4,3,6,5,2] => [1,5,4,3,6,2] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
 => ? = 4 - 3
[1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 5 - 3
[1,4,6,2,5,3] => [1,6,5,4,2,3] => [1,2,6,3,5,4] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 4 - 3
[1,4,6,3,2,5] => [1,3,6,4,2,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 1 = 4 - 3
[1,4,6,3,5,2] => [1,5,6,4,3,2] => [1,2,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 4 - 3
[1,5,3,2,4,6] => [1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[1,5,3,2,6,4] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[1,5,3,6,2,4] => [1,6,5,2,3,4] => [1,2,6,4,3,5] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
 => ? = 4 - 3
[1,5,6,3,2,4] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[1,6,2,4,3,5] => [1,6,4,2,3,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 4 - 3
[1,6,3,2,4,5] => [1,3,6,2,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 1 = 4 - 3
[1,6,3,2,5,4] => [1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[1,6,3,4,2,5] => [1,4,6,3,2,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
 => ? = 5 - 3
[1,6,3,5,2,4] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 4 - 3
[1,6,4,2,3,5] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 5 - 3
[1,6,4,2,5,3] => [1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
 => ? = 4 - 3
[1,6,4,3,2,5] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[1,6,4,3,5,2] => [1,5,4,6,3,2] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
 => ? = 4 - 3
[1,6,5,3,2,4] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,4,6,3,5] => [2,1,6,4,3,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 1 = 4 - 3
[2,1,4,6,5,3] => [2,1,5,4,6,3] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,5,3,4,6] => [2,1,5,3,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,5,3,6,4] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,5,4,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[2,1,5,4,6,3] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,5,6,3,4] => [2,1,6,3,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,5,6,4,3] => [2,1,4,6,5,3] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[2,1,6,4,3,5] => [2,1,4,6,3,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[2,1,6,4,5,3] => [2,1,5,6,4,3] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,6,5,3,4] => [2,1,5,3,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,1,6,5,4,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[2,3,1,5,4,6] => [3,2,1,5,4,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,3,1,5,6,4] => [3,2,1,6,5,4] => [1,3,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
 => ? = 4 - 3
[2,3,1,6,5,4] => [3,2,1,5,6,4] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 4 - 3
[2,3,6,1,5,4] => [6,2,3,5,1,4] => [1,6,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(1,10),(1,15),(2,8),(2,11),(2,15),(3,7),(3,8),(3,9),(4,7),(4,10),(4,11),(4,15),(5,17),(7,12),(7,13),(7,16),(8,13),(8,16),(9,12),(9,16),(10,12),(10,14),(11,5),(11,13),(11,14),(12,18),(13,17),(13,18),(14,17),(14,18),(15,5),(15,14),(15,16),(16,17),(16,18),(17,6),(18,6)],19)
 => ? = 4 - 3
[3,2,1,5,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[3,2,1,5,6,4] => [2,3,1,6,5,4] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 1 = 4 - 3
[3,2,1,6,5,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 4 - 3
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,6,5,4,7] => [1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,3,2,7,6,5,4] => [1,3,2,5,6,7,4] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3
[1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 4 - 3

Description

The number of minimal elements in a poset.

Matching statistic: St001851
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001851: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%

Values

[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => 0 = 4 - 4
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => 0 = 4 - 4
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => 0 = 4 - 4
[1,5,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => 0 = 4 - 4
[2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 0 = 4 - 4
[2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 0 = 4 - 4
[2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 0 = 4 - 4
[2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 0 = 4 - 4
[5,1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => 0 = 4 - 4
[5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 0 = 4 - 4
[1,2,4,3,5,6] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 0 = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 0 = 4 - 4
[1,2,4,6,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 0 = 4 - 4
[1,2,6,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 0 = 4 - 4
[1,3,2,4,5,6] => [1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => 0 = 4 - 4
[1,3,2,4,6,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => 0 = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 0 = 4 - 4
[1,3,2,5,6,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 0 = 4 - 4
[1,3,2,6,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => 0 = 4 - 4
[1,3,2,6,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 0 = 4 - 4
[1,3,4,2,5,6] => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1 = 5 - 4
[1,3,4,2,6,5] => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1 = 5 - 4
[1,3,4,6,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1 = 5 - 4
[1,3,5,2,4,6] => [1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => 0 = 4 - 4
[1,3,5,2,6,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => 0 = 4 - 4
[1,3,5,6,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => 0 = 4 - 4
[1,3,6,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => 0 = 4 - 4
[1,3,6,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 0 = 4 - 4
[1,3,6,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1 = 5 - 4
[1,3,6,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => 0 = 4 - 4
[1,4,2,3,5,6] => [1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => ? = 5 - 4
[1,4,2,3,6,5] => [1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => ? = 5 - 4
[1,4,2,5,3,6] => [1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => 0 = 4 - 4
[1,4,2,5,6,3] => [1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => 0 = 4 - 4
[1,4,2,6,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => ? = 5 - 4
[1,4,2,6,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => 0 = 4 - 4
[1,4,3,2,5,6] => [1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => 0 = 4 - 4
[1,4,3,2,6,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => 0 = 4 - 4
[1,4,3,5,2,6] => [1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => 0 = 4 - 4
[1,4,3,5,6,2] => [1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => 0 = 4 - 4
[1,4,3,6,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => 0 = 4 - 4
[1,4,3,6,5,2] => [1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => 0 = 4 - 4
[1,4,6,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => ? = 5 - 4
[1,4,6,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => 0 = 4 - 4
[1,4,6,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => 0 = 4 - 4
[1,4,6,3,5,2] => [1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => 0 = 4 - 4
[1,5,3,2,4,6] => [1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => 0 = 4 - 4
[1,5,3,2,6,4] => [1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => 0 = 4 - 4
[1,5,3,6,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => 0 = 4 - 4
[1,5,6,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => 0 = 4 - 4
[1,6,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 0 = 4 - 4
[1,6,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => 0 = 4 - 4
[1,6,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 0 = 4 - 4
[1,6,3,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1 = 5 - 4
[1,6,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => ? = 5 - 4
[2,1,4,3,5,6] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => ? = 4 - 4
[2,1,4,5,3,6] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => ? = 4 - 4
[2,1,4,5,6,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => ? = 4 - 4
[2,1,4,6,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => ? = 4 - 4
[2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => ? = 4 - 4
[2,1,5,3,4,6] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => ? = 4 - 4
[2,1,5,3,6,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => ? = 4 - 4
[2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => ? = 4 - 4
[2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => ? = 4 - 4
[2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => ? = 4 - 4
[2,1,5,6,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => ? = 4 - 4
[2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => ? = 4 - 4
[2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => ? = 4 - 4
[2,1,6,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => ? = 4 - 4
[2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => ? = 4 - 4
[2,3,1,5,4,6] => [2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => ? = 4 - 4
[2,3,1,5,6,4] => [2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => ? = 4 - 4
[2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => ? = 4 - 4
[2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => ? = 4 - 4
[2,4,1,3,5,6] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => ? = 4 - 4
[2,4,1,3,6,5] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => ? = 4 - 4
[2,4,1,5,3,6] => [2,4,1,5,3] => [2,4,1,5,3] => [5,-2,1,-4,3] => ? = 5 - 4
[2,4,1,5,6,3] => [2,4,1,5,3] => [2,4,1,5,3] => [5,-2,1,-4,3] => ? = 5 - 4
[2,4,1,6,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => ? = 4 - 4
[2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [5,-2,1,-4,3] => ? = 5 - 4
[2,4,3,1,5,6] => [2,4,3,1,5] => [2,4,3,1,5] => [3,-2,-4,1,5] => ? = 4 - 4
[2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => [3,-2,-4,1,5] => ? = 4 - 4
[2,4,3,5,1,6] => [2,4,3,5,1] => [2,4,3,5,1] => [3,-2,5,-4,1] => ? = 4 - 4
[2,4,3,5,6,1] => [2,4,3,5,1] => [2,4,3,5,1] => [3,-2,5,-4,1] => ? = 4 - 4
[2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => [3,-2,-4,1,5] => ? = 4 - 4
[2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [3,-2,5,-4,1] => ? = 4 - 4
[2,4,6,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => ? = 4 - 4
[2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [5,-2,1,-4,3] => ? = 5 - 4
[2,4,6,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => [3,-2,-4,1,5] => ? = 4 - 4
[2,4,6,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [3,-2,5,-4,1] => ? = 4 - 4
[2,5,1,4,3,6] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => ? = 4 - 4
[2,5,1,4,6,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => ? = 4 - 4
[2,5,1,6,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => ? = 4 - 4
[2,5,6,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => ? = 4 - 4
[2,6,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => ? = 4 - 4
[2,6,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => ? = 4 - 4
[2,6,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => ? = 4 - 4
[2,6,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => ? = 4 - 4
[2,6,3,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => ? = 4 - 4

Description

The number of Hecke atoms of a signed permutation. For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set $$ \{ w\in\mathfrak H_n | w^{-1} \star w = z\}, $$ where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.

The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001260The permanent of an alternating sign matrix. St001568The smallest positive integer that does not appear twice in the partition. St001645The pebbling number of a connected graph. St000787The number of flips required to make a perfect matching noncrossing. St001964The interval resolution global dimension of a poset. 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. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001434The number of negative sum pairs of a signed permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001116The game chromatic number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001861The number of Bruhat lower covers of a permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001624The breadth of a lattice. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000069The number of maximal elements of a poset. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St001625The Möbius invariant of a lattice. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St001344The neighbouring number of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001381The fertility of a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001947The number of ties in a parking function. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation.