Identifier
Values
[1,0,1,0] => [3,1,2] => [1,3,2] => ([(1,2)],3) => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 0
[1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4) => 0
[1,1,0,0,1,0] => [2,4,1,3] => [1,2,4,3] => ([(2,3)],4) => 0
[1,1,0,1,0,0] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4) => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5) => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,5,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => ([(4,5)],6) => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ([(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) => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,6,7,5,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,7,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [1,6,7,4,5,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,7,6,3,4,5,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [1,6,7,3,4,5,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,7,3,4,5,6,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => ([(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
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [1,5,6,3,2,7,4] => ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [1,2,7,6,5,3,4] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [1,7,6,5,2,3,4] => ([(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
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => [1,6,7,5,2,3,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [1,2,3,7,6,4,5] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [1,2,7,6,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [1,7,6,2,3,4,5] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [1,6,7,2,3,4,5] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [1,2,3,4,5,7,6] => ([(5,6)],7) => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7) => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
>>> Load all 102 entries. <<<
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [1,7,2,3,4,5,6] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
search for individual values
searching the database for the individual values of this statistic
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.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.