Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000259
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000456
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 34%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 - 1
[2,1] => [1,2] => ([],2)
 => ([],2)
 => ? = 0 - 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? = 1 - 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? = 1 - 1
[3,2,1] => [1,2,3] => ([],3)
 => ([],3)
 => ? = 0 - 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 1 - 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 2 - 1
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? = 1 - 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 1 - 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? = 1 - 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? = 1 - 1
[4,3,2,1] => [1,2,3,4] => ([],4)
 => ([],4)
 => ? = 0 - 1
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 3 - 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 - 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 3 - 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,2,5,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 1 - 1
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[3,4,2,5,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 2 - 1
[3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 1 - 1
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 2 - 1
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1 - 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[3,5,4,2,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1 - 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[4,2,3,5,1] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 2 - 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1 - 1
[4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
[4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[4,3,5,2,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1 - 1
[4,5,1,2,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 2 - 1
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 - 1
[4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 - 1
[4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ? = 1 - 1
[5,1,2,3,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[5,1,2,4,3] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[5,1,3,2,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[5,1,3,4,2] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[5,1,4,2,3] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[5,1,4,3,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 - 1
[5,2,1,3,4] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 - 1
[2,3,4,6,1,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,3,5,6,1,4] => [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)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,3,6,4,1,5] => [5,1,4,6,3,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,3,6,5,1,4] => [4,1,5,6,3,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,4,3,6,1,5] => [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)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,4,5,6,1,3] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,4,6,3,1,5] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,4,6,5,1,3] => [3,1,5,6,4,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,5,3,6,1,4] => [4,1,6,3,5,2] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,5,4,6,1,3] => [3,1,6,4,5,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,5,6,1,3,4] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,5,6,1,4,3] => [3,4,1,6,5,2] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,5,6,3,1,4] => [4,1,3,6,5,2] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,5,6,4,1,3] => [3,1,4,6,5,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
[2,6,1,3,4,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,1,3,5,4] => [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,1,4,3,5] => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,1,4,5,3] => [3,5,4,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,1,5,3,4] => [4,3,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,1,5,4,3] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,3,1,4,5] => [5,4,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,3,1,5,4] => [4,5,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,3,4,1,5] => [5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,6,3,5,1,4] => [4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,6,4,1,3,5] => [5,3,1,4,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,4,1,5,3] => [3,5,1,4,6,2] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,6,4,3,1,5] => [5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,6,4,5,1,3] => [3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 2 - 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001632
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0 - 2
[2,1] => ([(0,1)],2)
 => ([],2)
 => ([],2)
 => ? = 0 - 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 1 - 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 1 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],3)
 => ? = 0 - 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 1 - 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => ([],2)
 => ? = 2 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 1 - 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 1 - 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 1 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 1 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],4)
 => ? = 0 - 2
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? = 1 - 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 3 - 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2 - 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4)],5)
 => ? = 1 - 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 3 - 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2 - 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2 - 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ? = 1 - 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 1 - 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? = 2 - 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? = 1 - 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 - 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1 - 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? = 2 - 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4)],5)
 => ? = 1 - 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 - 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 2
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2 - 2
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,5,6,3,1,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,6,3,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,6,4,3,1,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,6,5,3,1,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[3,6,4,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[3,6,4,2,1,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[4,1,3,5,6,2] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[4,5,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[5,1,2,4,6,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[5,1,4,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[5,1,4,6,3,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[5,4,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[2,3,4,7,5,1,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,3,5,7,4,1,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,3,6,7,4,1,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,3,7,4,1,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,3,7,4,5,1,6] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,3,7,5,4,1,6] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,3,7,6,4,1,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[2,4,5,7,3,1,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,4,6,7,3,1,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,4,7,3,1,5,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,5,6,7,3,1,4] => ([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,5,7,3,4,1,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,6,7,3,1,4,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,6,7,3,4,1,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,6,7,4,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,6,7,4,3,1,5] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,6,7,5,3,1,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
[2,7,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,7,3,4,1,5,6] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,7,3,4,5,1,6] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 2 - 2
[2,7,4,1,3,5,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,7,4,3,1,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,7,4,5,3,1,6] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,7,5,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => 0 = 2 - 2
[2,7,5,3,4,1,6] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 2 - 2
[2,7,5,6,3,1,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 2 - 2
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000455
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
 => ([(0,1)],2)
 => -1 = 0 - 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 0 - 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 0 - 1
[2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,3,4,6,5,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,3,5,6,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,3,6,5,4,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,4,5,6,3,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,4,6,5,3,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,5,6,4,3,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,6,5,4,3,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[3,4,5,6,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[3,4,6,5,2,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[3,5,6,4,2,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[3,6,5,4,2,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[4,5,6,3,2,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[4,6,5,3,2,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[5,6,4,3,2,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => -1 = 0 - 1
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.