- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1957 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

Demazure product with inverse

Description

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.