- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Map

LLPS

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.

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.