- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1609 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

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.