- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤ

Values

[.,.] => [1] => [1] => 0

[.,[.,.]] => [2,1] => [1,1] => 0

[[.,.],.] => [1,2] => [2] => 0

[.,[.,[.,.]]] => [3,2,1] => [1,1,1] => 0

[.,[[.,.],.]] => [2,3,1] => [2,1] => 0

[[.,.],[.,.]] => [3,1,2] => [2,1] => 0

[[.,[.,.]],.] => [2,1,3] => [2,1] => 0

[[[.,.],.],.] => [1,2,3] => [3] => 0

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1] => 0

[.,[.,[[.,.],.]]] => [3,4,2,1] => [2,1,1] => 0

[.,[[.,.],[.,.]]] => [4,2,3,1] => [2,1,1] => 0

[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,1] => 0

[.,[[[.,.],.],.]] => [2,3,4,1] => [3,1] => 0

[[.,.],[.,[.,.]]] => [4,3,1,2] => [2,1,1] => 0

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

[[.,[.,.]],[.,.]] => [4,2,1,3] => [2,1,1] => 0

[[[.,.],.],[.,.]] => [4,1,2,3] => [3,1] => 0

[[.,[.,[.,.]]],.] => [3,2,1,4] => [2,1,1] => 0

[[.,[[.,.],.]],.] => [2,3,1,4] => [3,1] => 0

[[[.,.],[.,.]],.] => [3,1,2,4] => [3,1] => 0

[[[.,[.,.]],.],.] => [2,1,3,4] => [3,1] => 0

[[[[.,.],.],.],.] => [1,2,3,4] => [4] => 0

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1] => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,1,1,1] => 0

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [2,1,1,1] => 0

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,1,1] => 0

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,1,1] => 0

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [2,1,1,1] => 0

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

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [2,1,1,1] => 0

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [3,1,1] => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [2,1,1,1] => 0

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,1,1] => 0

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [3,1,1] => 0

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,1,1] => 0

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [4,1] => 0

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [2,1,1,1] => 0

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

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

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

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

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [2,1,1,1] => 0

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

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [3,1,1] => 0

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

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [2,1,1,1] => 0

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,1,1] => 0

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [3,1,1] => 0

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [3,1,1] => 0

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [4,1] => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [2,1,1,1] => 0

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,1,1] => 0

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [3,1,1] => 0

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,1,1] => 0

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,1] => 0

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [3,1,1] => 0

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

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [3,1,1] => 0

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [4,1] => 0

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,1,1] => 0

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [4,1] => 0

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [4,1] => 0

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [4,1] => 0

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5] => 0

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,1,1,1] => 0

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [2,1,1,1,1] => 0

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,1,1,1,1] => 0

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [3,1,1,1] => 0

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [2,1,1,1,1] => 0

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

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [2,1,1,1,1] => 0

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [3,1,1,1] => 0

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [2,1,1,1,1] => 0

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,1,1,1] => 0

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [3,1,1,1] => 0

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [3,1,1,1] => 0

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,1,1] => 0

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [2,1,1,1,1] => 0

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

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

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

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

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [2,1,1,1,1] => 0

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

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [3,1,1,1] => 0

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

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [2,1,1,1,1] => 0

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [3,1,1,1] => 0

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [3,1,1,1] => 0

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [3,1,1,1] => 0

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [4,1,1] => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,1,1,1,1] => 0

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [3,1,1,1] => 0

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [3,1,1,1] => 0

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [3,1,1,1] => 0

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,1,1] => 0

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [3,1,1,1] => 0

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

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [3,1,1,1] => 0

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [4,1,1] => 0

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Description

The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

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.