- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000183: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.

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.