- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 189 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2\ q$

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 graph

Description

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

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

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.