- FindStat

Identifier

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.

Map

Loehr-Warrington

Description

Return a partition whose diagonal inversion number is the length of the preimage.

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.