- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001382: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1852 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of boxes in the diagram of a partition that do not lie in its Durfee square.

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.