Identifier
Values
[.,[.,[.,.]]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[.,[[.,.],[.,.]]] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4) => 0
[[.,.],[.,[.,.]]] => [4,3,1,2] => [1,3] => ([(2,3)],4) => 0
[[.,[.,.]],[.,.]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5) => 0
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [1,4] => ([(3,4)],5) => 0
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [1,4] => ([(3,4)],5) => 0
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [1,4] => ([(3,4)],5) => 0
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [1,4] => ([(3,4)],5) => 0
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5) => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,4] => ([(3,4)],5) => 0
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [2,4] => ([(3,5),(4,5)],6) => 0
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => [1,5] => ([(4,5)],6) => 0
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => [1,5] => ([(4,5)],6) => 0
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => [1,5] => ([(4,5)],6) => 0
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => [2,4] => ([(3,5),(4,5)],6) => 0
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => [1,5] => ([(4,5)],6) => 0
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => [1,5] => ([(4,5)],6) => 0
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => [1,5] => ([(4,5)],6) => 0
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => [1,5] => ([(4,5)],6) => 0
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => [1,5] => ([(4,5)],6) => 0
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => [1,5] => ([(4,5)],6) => 0
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6) => 0
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => [1,5] => ([(4,5)],6) => 0
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [1,5] => ([(4,5)],6) => 0
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => [1,5] => ([(4,5)],6) => 0
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => [1,5] => ([(4,5)],6) => 0
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => [1,5] => ([(4,5)],6) => 0
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => 0
[.,[.,[.,[.,[[[.,.],.],.]]]]] => [5,6,7,4,3,2,1] => [4,1,1,1] => ([(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
[.,[.,[.,[[[[.,.],.],.],.]]]] => [4,5,6,7,3,2,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[.,[[.,.],[[[.,.],.],.]]]] => [5,6,7,3,4,2,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[.,[[[.,.],.],[[.,.],.]]]] => [6,7,3,4,5,2,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[.,[[[[[.,.],.],.],.],.]]] => [3,4,5,6,7,2,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,.],[[[[.,.],.],.],.]]] => [4,5,6,7,2,3,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[.,.],.],[[[.,.],.],.]]] => [5,6,7,2,3,4,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[.,.],.],.],[[.,.],.]]] => [6,7,2,3,4,5,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,.],.],.],.],[.,.]]] => [7,2,3,4,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,[.,[[[.,.],.],.]]],.]] => [4,5,6,3,2,7,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[[.,[[.,.],[[.,.],.]]],.]] => [5,6,3,4,2,7,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[[.,[[[[.,.],.],.],.]],.]] => [3,4,5,6,2,7,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[.,.],[[[.,.],.],.]],.]] => [4,5,6,2,3,7,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[.,.],.],[[.,.],.]],.]] => [5,6,2,3,4,7,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,.],.],.],[.,.]],.]] => [6,2,3,4,5,7,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[.,[.,[[.,.],.]]],.],.]] => [4,5,3,2,6,7,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[.,[[[.,[[[.,.],.],.]],.],.]] => [3,4,5,2,6,7,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
>>> Load all 213 entries. <<<
[.,[[[[.,.],[[.,.],.]],.],.]] => [4,5,2,3,6,7,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,.],.],[.,.]],.],.]] => [5,2,3,4,6,7,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[.,[[.,.],.]],.],.],.]] => [3,4,2,5,6,7,1] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,.],[.,.]],.],.],.]] => [4,2,3,5,6,7,1] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,[.,.]],.],.],.],.]] => [3,2,4,5,6,7,1] => [2,5] => ([(4,6),(5,6)],7) => 0
[[.,.],[.,[.,[[[.,.],.],.]]]] => [5,6,7,4,3,1,2] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,.],[.,[[[[.,.],.],.],.]]] => [4,5,6,7,3,1,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[.,.],[[[.,.],.],.]]] => [5,6,7,3,4,1,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[[.,.],.],[[.,.],.]]] => [6,7,3,4,5,1,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[[[.,.],.],.],[.,.]]] => [7,3,4,5,6,1,2] => [1,6] => ([(5,6)],7) => 0
[[.,.],[[.,[[[.,.],.],.]],.]] => [4,5,6,3,7,1,2] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[[.,.],[[.,.],.]],.]] => [5,6,3,4,7,1,2] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[[[.,.],.],[.,.]],.]] => [6,3,4,5,7,1,2] => [1,6] => ([(5,6)],7) => 0
[[.,.],[[[.,[[.,.],.]],.],.]] => [4,5,3,6,7,1,2] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[.,.],[[[[.,.],[.,.]],.],.]] => [5,3,4,6,7,1,2] => [1,6] => ([(5,6)],7) => 0
[[.,.],[[[[.,[.,.]],.],.],.]] => [4,3,5,6,7,1,2] => [1,6] => ([(5,6)],7) => 0
[[.,[.,.]],[.,[[[.,.],.],.]]] => [5,6,7,4,2,1,3] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[.,.]],[[[[.,.],.],.],.]] => [4,5,6,7,2,1,3] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],.],[.,[[[.,.],.],.]]] => [5,6,7,4,1,2,3] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],.],[[.,.],[.,[.,.]]]] => [7,6,4,5,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[.,.],.],[[.,.],[[.,.],.]]] => [6,7,4,5,1,2,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[.,.],.],[[.,[.,.]],[.,.]]] => [7,5,4,6,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[.,.],.],[[[.,.],.],[.,.]]] => [7,4,5,6,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[[.,.],.],[[.,[.,[.,.]]],.]] => [6,5,4,7,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[.,.],.],[[.,[[.,.],.]],.]] => [5,6,4,7,1,2,3] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[.,.],.],[[[.,.],[.,.]],.]] => [6,4,5,7,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[[.,.],.],[[[.,[.,.]],.],.]] => [5,4,6,7,1,2,3] => [1,6] => ([(5,6)],7) => 0
[[.,[.,[.,.]]],[[[.,.],.],.]] => [5,6,7,3,2,1,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[[.,.],.]],[[[.,.],.],.]] => [5,6,7,2,3,1,4] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[.,.]],[[[.,.],.],.]] => [5,6,7,3,1,2,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[.,.]],.],[[[.,.],.],.]] => [5,6,7,2,1,3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],.],.],[.,[.,[.,.]]]] => [7,6,5,1,2,3,4] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[[.,.],.],.],[.,[[.,.],.]]] => [6,7,5,1,2,3,4] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[.,.],.],.],[[.,.],[.,.]]] => [7,5,6,1,2,3,4] => [1,6] => ([(5,6)],7) => 0
[[[[.,.],.],.],[[.,[.,.]],.]] => [6,5,7,1,2,3,4] => [1,6] => ([(5,6)],7) => 0
[[.,[.,[[.,.],.]]],[[.,.],.]] => [6,7,3,4,2,1,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[[[.,.],.],.]],[[.,.],.]] => [6,7,2,3,4,1,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[[.,.],.]],[[.,.],.]] => [6,7,3,4,1,2,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],.],[.,.]],[.,[.,.]]] => [7,6,4,1,2,3,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[[.,.],.],[.,.]],[[.,.],.]] => [6,7,4,1,2,3,5] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[.,[.,[.,.]]],.],[[.,.],.]] => [6,7,3,2,1,4,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[[.,[[.,.],.]],.],[[.,.],.]] => [6,7,2,3,1,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],[.,.]],.],[[.,.],.]] => [6,7,3,1,2,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[.,[.,.]],.],.],[[.,.],.]] => [6,7,2,1,3,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],.],.],.],[.,[.,.]]] => [7,6,1,2,3,4,5] => [1,6] => ([(5,6)],7) => 0
[[.,[[[[.,.],.],.],.]],[.,.]] => [7,2,3,4,5,1,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[[[.,.],.],.]],[.,.]] => [7,3,4,5,1,2,6] => [1,6] => ([(5,6)],7) => 0
[[[[.,.],.],[.,[.,.]]],[.,.]] => [7,5,4,1,2,3,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[[.,.],.],[[.,.],.]],[.,.]] => [7,4,5,1,2,3,6] => [1,6] => ([(5,6)],7) => 0
[[[[[.,.],.],.],[.,.]],[.,.]] => [7,5,1,2,3,4,6] => [1,6] => ([(5,6)],7) => 0
[[[.,[[[.,.],.],.]],.],[.,.]] => [7,2,3,4,1,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],[[.,.],.]],.],[.,.]] => [7,3,4,1,2,5,6] => [1,6] => ([(5,6)],7) => 0
[[[[[.,.],.],[.,.]],.],[.,.]] => [7,4,1,2,3,5,6] => [1,6] => ([(5,6)],7) => 0
[[[[.,[[.,.],.]],.],.],[.,.]] => [7,2,3,1,4,5,6] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],[.,.]],.],.],[.,.]] => [7,3,1,2,4,5,6] => [1,6] => ([(5,6)],7) => 0
[[[[[.,[.,.]],.],.],.],[.,.]] => [7,2,1,3,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => 0
[[.,[.,[.,[[[.,.],.],.]]]],.] => [4,5,6,3,2,1,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[.,[[.,.],[[.,.],.]]]],.] => [5,6,3,4,2,1,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[.,[[[[.,.],.],.],.]]],.] => [3,4,5,6,2,1,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[.,.],[[[.,.],.],.]]],.] => [4,5,6,2,3,1,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[.,.],.],[[.,.],.]]],.] => [5,6,2,3,4,1,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[[.,.],.],.],[.,.]]],.] => [6,2,3,4,5,1,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[.,[.,[[.,.],.]]],.]],.] => [4,5,3,2,6,1,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[.,[[.,[[[.,.],.],.]],.]],.] => [3,4,5,2,6,1,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[.,.],[[.,.],.]],.]],.] => [4,5,2,3,6,1,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[[.,.],.],[.,.]],.]],.] => [5,2,3,4,6,1,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[.,[[.,.],.]],.],.]],.] => [3,4,2,5,6,1,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[.,[[[[.,.],[.,.]],.],.]],.] => [4,2,3,5,6,1,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[.,[[[[.,[.,.]],.],.],.]],.] => [3,2,4,5,6,1,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[.,.],[.,[[[.,.],.],.]]],.] => [4,5,6,3,1,2,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[[.,.],[[.,.],.]]],.] => [5,6,3,4,1,2,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[[[.,.],.],[.,.]]],.] => [6,3,4,5,1,2,7] => [1,6] => ([(5,6)],7) => 0
[[[.,.],[[.,[[.,.],.]],.]],.] => [4,5,3,6,1,2,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[.,.],[[[.,.],[.,.]],.]],.] => [5,3,4,6,1,2,7] => [1,6] => ([(5,6)],7) => 0
[[[.,.],[[[.,[.,.]],.],.]],.] => [4,3,5,6,1,2,7] => [1,6] => ([(5,6)],7) => 0
[[[.,[.,.]],[[[.,.],.],.]],.] => [4,5,6,2,1,3,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],.],[.,[.,[.,.]]]],.] => [6,5,4,1,2,3,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[[[[.,.],.],[.,[[.,.],.]]],.] => [5,6,4,1,2,3,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[.,.],.],[[.,.],[.,.]]],.] => [6,4,5,1,2,3,7] => [1,6] => ([(5,6)],7) => 0
[[[[.,.],.],[[.,[.,.]],.]],.] => [5,4,6,1,2,3,7] => [1,6] => ([(5,6)],7) => 0
[[[.,[.,[.,.]]],[[.,.],.]],.] => [5,6,3,2,1,4,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[[.,[[.,.],.]],[[.,.],.]],.] => [5,6,2,3,1,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],[.,.]],[[.,.],.]],.] => [5,6,3,1,2,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[.,[.,.]],.],[[.,.],.]],.] => [5,6,2,1,3,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],.],.],[.,[.,.]]],.] => [6,5,1,2,3,4,7] => [1,6] => ([(5,6)],7) => 0
[[[.,[[[.,.],.],.]],[.,.]],.] => [6,2,3,4,1,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],[[.,.],.]],[.,.]],.] => [6,3,4,1,2,5,7] => [1,6] => ([(5,6)],7) => 0
[[[[[.,.],.],[.,.]],[.,.]],.] => [6,4,1,2,3,5,7] => [1,6] => ([(5,6)],7) => 0
[[[[.,[[.,.],.]],.],[.,.]],.] => [6,2,3,1,4,5,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],[.,.]],.],[.,.]],.] => [6,3,1,2,4,5,7] => [1,6] => ([(5,6)],7) => 0
[[[[[.,[.,.]],.],.],[.,.]],.] => [6,2,1,3,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[.,[.,[.,[[.,.],.]]]],.],.] => [4,5,3,2,1,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[[[.,[.,[[[.,.],.],.]]],.],.] => [3,4,5,2,1,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[[.,.],[[.,.],.]]],.],.] => [4,5,2,3,1,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[[[.,.],.],[.,.]]],.],.] => [5,2,3,4,1,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[[.,[[.,.],.]],.]],.],.] => [3,4,2,5,1,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[.,[[[.,.],[.,.]],.]],.],.] => [4,2,3,5,1,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[.,[[[.,[.,.]],.],.]],.],.] => [3,2,4,5,1,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[.,.],[.,[[.,.],.]]],.],.] => [4,5,3,1,2,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[.,.],[[.,.],[.,.]]],.],.] => [5,3,4,1,2,6,7] => [1,6] => ([(5,6)],7) => 0
[[[[.,.],[[.,[.,.]],.]],.],.] => [4,3,5,1,2,6,7] => [1,6] => ([(5,6)],7) => 0
[[[[.,[.,.]],[[.,.],.]],.],.] => [4,5,2,1,3,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],.],[.,[.,.]]],.],.] => [5,4,1,2,3,6,7] => [1,6] => ([(5,6)],7) => 0
[[[[.,[[.,.],.]],[.,.]],.],.] => [5,2,3,1,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[[.,.],[.,.]],[.,.]],.],.] => [5,3,1,2,4,6,7] => [1,6] => ([(5,6)],7) => 0
[[[[[.,[.,.]],.],[.,.]],.],.] => [5,2,1,3,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[.,[.,[[.,.],.]]],.],.],.] => [3,4,2,1,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[.,[[.,.],[.,.]]],.],.],.] => [4,2,3,1,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[[[[.,[[.,[.,.]],.]],.],.],.] => [3,2,4,1,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[[.,.],[.,[.,.]]],.],.],.] => [4,3,1,2,5,6,7] => [1,6] => ([(5,6)],7) => 0
[[[[[.,[.,.]],[.,.]],.],.],.] => [4,2,1,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => 0
[[[[[.,[.,[.,.]]],.],.],.],.] => [3,2,1,4,5,6,7] => [2,5] => ([(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
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
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.