- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 9959 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

toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.