- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1902 entries. <<<

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

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].