- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2050 entries. <<<

Map

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this Sylvester class.

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.