- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000308: Permutations ⟶ ℤ

Values

[.,.] => [1] => [1] => [1] => 1

[.,[.,.]] => [2,1] => [1,2] => [1,2] => 2

[[.,.],.] => [1,2] => [1,2] => [1,2] => 2

[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => [1,2,3] => 3

[.,[[.,.],.]] => [2,3,1] => [1,2,3] => [1,2,3] => 3

[[.,.],[.,.]] => [3,1,2] => [1,2,3] => [1,2,3] => 3

[[.,[.,.]],.] => [2,1,3] => [1,3,2] => [1,3,2] => 2

[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 3

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4

[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4

[.,[[.,.],[.,.]]] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4

[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3

[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4

[[.,.],[.,[.,.]]] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 4

[[.,.],[[.,.],.]] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4

[[.,[.,.]],[.,.]] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3

[[[.,.],.],[.,.]] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4

[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3

[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 3

[[[.,.],[.,.]],.] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3

[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 3

[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 4

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 4

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 4

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,2,4,3] => [1,3,5,4,2] => 3

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 4

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 4

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 260 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

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searching the database for statistics with the same generating function

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = q^{2} + 4\ q^{3}$

$F_{4} = 6\ q^{3} + 8\ q^{4}$

$F_{5} = 6\ q^{3} + 20\ q^{4} + 16\ q^{5}$

$F_{6} = 4\ q^{3} + 40\ q^{4} + 56\ q^{5} + 32\ q^{6}$

Description

The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

inverse

Description

Sends a permutation to its inverse.

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.