- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 11024 entries. <<<

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

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.