- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

Robinson-Schensted insertion tableau

Description

Sends a permutation to its Robinson-Schensted insertion tableau.
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 its corresponding insertion tableau.