- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

[1,2] => [.,[.,.]] => ([(0,1)],2) => ([(0,1)],2) => -1

[2,1] => [[.,.],.] => ([(0,1)],2) => ([(0,1)],2) => -1

[1,2,3] => [.,[.,[.,.]]] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

[1,3,2] => [.,[[.,.],.]] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

[2,1,3] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

[2,3,1] => [[.,[.,.]],.] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

[3,1,2] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

[3,2,1] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 graph

Description

Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.

Map

connected complement

Description

The componentwise connected complement of a graph.
For a connected graph $G$, this map returns the complement of $G$ if it is connected, otherwise $G$ itself. If $G$ is not connected, the map is applied to each connected component separately.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.