Processing math: 100%

Identifier
Values
[1] => {{1}} => [1] => ([],1) => 0
[1,2] => {{1},{2}} => [1,1] => ([(0,1)],2) => 1
[2,1] => {{1,2}} => [2] => ([],2) => 0
[1,2,3] => {{1},{2},{3}} => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
[1,3,2] => {{1},{2,3}} => [1,2] => ([(1,2)],3) => 1
[2,3,1] => {{1,2,3}} => [3] => ([],3) => 0
[1,2,3,4] => {{1},{2},{3},{4}} => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[1,2,4,3] => {{1},{2},{3,4}} => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[1,3,4,2] => {{1},{2,3,4}} => [1,3] => ([(2,3)],4) => 1
[2,3,4,1] => {{1,2,3,4}} => [4] => ([],4) => 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [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) => 4
[1,2,3,5,4] => {{1},{2},{3},{4,5}} => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,2,4,5,3] => {{1},{2},{3,4,5}} => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[1,3,4,5,2] => {{1},{2,3,4,5}} => [1,4] => ([(3,4)],5) => 1
[2,3,1,4,5] => {{1,2,3},{4},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,3,4,1,5] => {{1,2,3,4},{5}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,5,1] => {{1,2,3,4,5}} => [5] => ([],5) => 0
[2,3,5,1,4] => {{1,2,3,5},{4}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,3,5,4,1] => {{1,2,3,5},{4}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,4,1,3,5] => {{1,2,4},{3},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,4,1,5,3] => {{1,2,4,5},{3}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,4,3,1,5] => {{1,2,4},{3},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,4,3,5,1] => {{1,2,4,5},{3}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[2,5,1,3,4] => {{1,2,5},{3},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,5,1,4,3] => {{1,2,5},{3},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,5,3,1,4] => {{1,2,5},{3},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,5,3,4,1] => {{1,2,5},{3},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,4,2,5] => {{1,3,4},{2},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,4,5,2] => {{1,3,4,5},{2}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,5,4,2] => {{1,3,5},{2},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,2,4,1,5] => {{1,3,4},{2},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,2,4,5,1] => {{1,3,4,5},{2}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,2,5,1,4] => {{1,3,5},{2},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,2,5,4,1] => {{1,3,5},{2},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,1,2,5,3] => {{1,4,5},{2},{3}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,1,3,5,2] => {{1,4,5},{2},{3}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,2,1,5,3] => {{1,4,5},{2},{3}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,2,3,5,1] => {{1,4,5},{2},{3}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => [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) => 5
[1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}} => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}} => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,2,4,5,6,3] => {{1},{2},{3,4,5,6}} => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,3,4,5,6,2] => {{1},{2,3,4,5,6}} => [1,5] => ([(4,5)],6) => 1
[2,3,1,4,6,5] => {{1,2,3},{4},{5,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,3,4,1,6,5] => {{1,2,3,4},{5,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,3,4,5,6,1] => {{1,2,3,4,5,6}} => [6] => ([],6) => 0
[2,3,5,6,1,4] => {{1,2,3,5},{4,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,3,5,6,4,1] => {{1,2,3,5},{4,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,3,6,5,1,4] => {{1,2,3,6},{4,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,3,6,5,4,1] => {{1,2,3,6},{4,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,4,1,3,6,5] => {{1,2,4},{3},{5,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,4,3,1,6,5] => {{1,2,4},{3},{5,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,4,5,6,1,3] => {{1,2,4,6},{3,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,4,5,6,3,1] => {{1,2,4,6},{3,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,4,6,5,1,3] => {{1,2,4,5},{3,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,4,6,5,3,1] => {{1,2,4,5},{3,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,5,1,6,3,4] => {{1,2,5},{3},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,5,1,6,4,3] => {{1,2,5},{3},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,5,3,6,1,4] => {{1,2,5},{3},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,5,3,6,4,1] => {{1,2,5},{3},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,5,4,1,6,3] => {{1,2,5,6},{3,4}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,5,4,3,6,1] => {{1,2,5,6},{3,4}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,6,1,5,4,3] => {{1,2,6},{3},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,6,3,5,1,4] => {{1,2,6},{3},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,6,3,5,4,1] => {{1,2,6},{3},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,4,2,6,5] => {{1,3,4},{2},{5,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,5,6,2,4] => {{1,3,5},{2},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,5,6,4,2] => {{1,3,5},{2},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,6,5,4,2] => {{1,3,6},{2},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,4,1,6,5] => {{1,3,4},{2},{5,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,5,6,1,4] => {{1,3,5},{2},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,5,6,4,1] => {{1,3,5},{2},{4,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,6,5,1,4] => {{1,3,6},{2},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,6,5,4,1] => {{1,3,6},{2},{4,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,4,5,1,6,2] => {{1,3,5,6},{2,4}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,4,5,2,6,1] => {{1,3,5,6},{2,4}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,5,4,6,1,2] => {{1,3,4,6},{2,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,5,4,6,2,1] => {{1,3,4,6},{2,5}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,6,4,5,1,2] => {{1,3,4,5},{2,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,6,4,5,2,1] => {{1,3,4,5},{2,6}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,1,5,6,2,3] => {{1,4,6},{2},{3,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,1,5,6,3,2] => {{1,4,6},{2},{3,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,1,6,5,2,3] => {{1,4,5},{2},{3,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,1,6,5,3,2] => {{1,4,5},{2},{3,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,5,6,1,3] => {{1,4,6},{2},{3,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,5,6,3,1] => {{1,4,6},{2},{3,5}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,6,5,1,3] => {{1,4,5},{2},{3,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2,6,5,3,1] => {{1,4,5},{2},{3,6}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,1,5,6,2] => {{1,4,5,6},{2,3}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,3,2,5,6,1] => {{1,4,5,6},{2,3}} => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,1,4,3,6,2] => {{1,5,6},{2},{3,4}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,4,1,6,3] => {{1,5,6},{2},{3,4}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,4,3,6,1] => {{1,5,6},{2},{3,4}} => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}} => [1,1,1,1,1,2] => ([(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) => 5
[1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}} => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}} => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
>>> Load all 196 entries. <<<
[1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}} => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 2
[1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}} => [1,6] => ([(5,6)],7) => 1
[2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,3,4,1,6,7,5] => {{1,2,3,4},{5,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}} => [7] => ([],7) => 0
[2,3,5,6,1,7,4] => {{1,2,3,5},{4,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,5,6,4,7,1] => {{1,2,3,5},{4,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,6,5,7,1,4] => {{1,2,3,6},{4,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,6,5,7,4,1] => {{1,2,3,6},{4,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,7,5,6,1,4] => {{1,2,3,7},{4,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,3,7,5,6,4,1] => {{1,2,3,7},{4,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,1,3,6,7,5] => {{1,2,4},{3},{5,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,4,3,1,6,7,5] => {{1,2,4},{3},{5,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,5,6,7,3,1] => {{1,2,4,6},{3,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,5,7,6,1,3] => {{1,2,4,7},{3,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,5,7,6,3,1] => {{1,2,4,7},{3,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,6,5,1,7,3] => {{1,2,4,5},{3,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,4,6,5,3,7,1] => {{1,2,4,5},{3,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,5,1,6,3,7,4] => {{1,2,5},{3},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,5,1,6,4,7,3] => {{1,2,5},{3},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,5,3,6,1,7,4] => {{1,2,5},{3},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,5,3,6,4,7,1] => {{1,2,5},{3},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,5,4,6,7,1,3] => {{1,2,5,7},{3,4,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,5,4,6,7,3,1] => {{1,2,5,7},{3,4,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,5,4,7,6,1,3] => {{1,2,5,6},{3,4,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,5,4,7,6,3,1] => {{1,2,5,6},{3,4,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,6,1,5,7,3,4] => {{1,2,6},{3},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,6,1,5,7,4,3] => {{1,2,6},{3},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,6,3,5,7,1,4] => {{1,2,6},{3},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,6,3,5,7,4,1] => {{1,2,6},{3},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,6,4,5,1,7,3] => {{1,2,6,7},{3,4,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,6,4,5,3,7,1] => {{1,2,6,7},{3,4,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[2,7,1,5,6,3,4] => {{1,2,7},{3},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,7,1,5,6,4,3] => {{1,2,7},{3},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,7,3,5,6,1,4] => {{1,2,7},{3},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,7,3,5,6,4,1] => {{1,2,7},{3},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,4,2,6,7,5] => {{1,3,4},{2},{5,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,5,6,2,7,4] => {{1,3,5},{2},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,5,6,4,7,2] => {{1,3,5},{2},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,6,5,7,2,4] => {{1,3,6},{2},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,6,5,7,4,2] => {{1,3,6},{2},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,7,5,6,2,4] => {{1,3,7},{2},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,7,5,6,4,2] => {{1,3,7},{2},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,4,1,6,7,5] => {{1,3,4},{2},{5,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,5,6,1,7,4] => {{1,3,5},{2},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,5,6,4,7,1] => {{1,3,5},{2},{4,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,6,5,7,1,4] => {{1,3,6},{2},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,6,5,7,4,1] => {{1,3,6},{2},{4,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,7,5,6,1,4] => {{1,3,7},{2},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,2,7,5,6,4,1] => {{1,3,7},{2},{4,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,4,5,6,7,1,2] => {{1,3,5,7},{2,4,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,4,5,6,7,2,1] => {{1,3,5,7},{2,4,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,4,5,7,6,1,2] => {{1,3,5,6},{2,4,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,4,5,7,6,2,1] => {{1,3,5,6},{2,4,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,4,6,5,1,7,2] => {{1,3,6,7},{2,4,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,4,6,5,2,7,1] => {{1,3,6,7},{2,4,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,5,4,6,7,1,2] => {{1,3,4,6},{2,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,5,4,6,7,2,1] => {{1,3,4,6},{2,5,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,5,4,7,6,1,2] => {{1,3,4,7},{2,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,5,4,7,6,2,1] => {{1,3,4,7},{2,5,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,6,4,5,1,7,2] => {{1,3,4,5},{2,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[3,6,4,5,2,7,1] => {{1,3,4,5},{2,6,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,1,5,6,7,2,3] => {{1,4,6},{2},{3,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,1,5,6,7,3,2] => {{1,4,6},{2},{3,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,1,5,7,6,2,3] => {{1,4,7},{2},{3,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,1,5,7,6,3,2] => {{1,4,7},{2},{3,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,1,6,5,2,7,3] => {{1,4,5},{2},{3,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,1,6,5,3,7,2] => {{1,4,5},{2},{3,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,5,6,7,1,3] => {{1,4,6},{2},{3,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,5,6,7,3,1] => {{1,4,6},{2},{3,5,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,5,7,6,1,3] => {{1,4,7},{2},{3,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,5,7,6,3,1] => {{1,4,7},{2},{3,5,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,6,5,1,7,3] => {{1,4,5},{2},{3,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,6,5,3,7,1] => {{1,4,5},{2},{3,6,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,3,5,6,1,7,2] => {{1,4,6,7},{2,3,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,3,5,6,2,7,1] => {{1,4,6,7},{2,3,5}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,3,6,5,7,1,2] => {{1,4,5,7},{2,3,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,3,6,5,7,2,1] => {{1,4,5,7},{2,3,6}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,3,7,5,6,1,2] => {{1,4,5,6},{2,3,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[4,3,7,5,6,2,1] => {{1,4,5,6},{2,3,7}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,1,4,6,7,2,3] => {{1,5,7},{2},{3,4,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,1,4,6,7,3,2] => {{1,5,7},{2},{3,4,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,1,4,7,6,2,3] => {{1,5,6},{2},{3,4,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,1,4,7,6,3,2] => {{1,5,6},{2},{3,4,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,2,4,6,7,1,3] => {{1,5,7},{2},{3,4,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,2,4,6,7,3,1] => {{1,5,7},{2},{3,4,6}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,2,4,7,6,1,3] => {{1,5,6},{2},{3,4,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,2,4,7,6,3,1] => {{1,5,6},{2},{3,4,7}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,3,4,1,6,7,2] => {{1,5,6,7},{2,3,4}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,3,4,2,6,7,1] => {{1,5,6,7},{2,3,4}} => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[6,1,4,5,2,7,3] => {{1,6,7},{2},{3,4,5}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[6,1,4,5,3,7,2] => {{1,6,7},{2},{3,4,5}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[6,2,4,5,1,7,3] => {{1,6,7},{2},{3,4,5}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[6,2,4,5,3,7,1] => {{1,6,7},{2},{3,4,5}} => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
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Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs (i,π(i)) with i≤π(i).
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of {1,2,…,n}, this is the integer composition of n obtained by sorting the blocks by their minimal element and then taking the block sizes.