- FindStat

Identifier

Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000099: Permutations ⟶ ℤ (values match St000023The number of inner peaks of a permutation.)

Values

{{1}} => {{1}} => [1] => [1] => 1

{{1,2}} => {{1,2}} => [2,1] => [2,1] => 1

{{1},{2}} => {{1},{2}} => [1,2] => [1,2] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1

{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2

{{1,2,3,4,6},{5}} => {{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => [6,2,4,1,3,5] => 2

{{1,2,3,4},{5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => [2,1,6,3,4,5] => 2

{{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2

{{1,2,3,5,6},{4}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [6,3,5,1,2,4] => 2

{{1,2,3,5},{4,6}} => {{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => [6,2,5,4,1,3] => 2

{{1,2,3,5},{4},{6}} => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => [1,6,3,5,2,4] => 3

{{1,2,3,6},{4,5}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => [6,2,5,1,4,3] => 3

{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,1,2,6,4,5] => 2

{{1,2,3},{4,5},{6}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => [1,3,2,6,4,5] => 3

{{1,2,3,6},{4},{5}} => {{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => [6,3,5,1,4,2] => 3

{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => [3,2,1,6,4,5] => 2

{{1,2,3},{4},{5,6}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => [2,1,3,6,4,5] => 2

{{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2

{{1,2,4,5,6},{3}} => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [4,6,1,2,3,5] => 2

{{1,2,4,5},{3,6}} => {{1,4,5},{2,3,6}} => [4,3,6,5,1,2] => [2,6,5,1,4,3] => 3

{{1,2,4,5},{3},{6}} => {{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => [1,4,6,2,3,5] => 2

{{1,2,4,6},{3,5}} => {{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => [2,5,1,3,6,4] => 3

{{1,2,4},{3,5,6}} => {{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [3,6,5,1,2,4] => 2

{{1,2,4},{3,5},{6}} => {{1},{2,4,5},{3,6}} => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 2

{{1,2,4,6},{3},{5}} => {{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => [4,6,2,5,1,3] => 3

{{1,2,4},{3,6},{5}} => {{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [2,3,6,1,4,5] => 2

{{1,2,4},{3},{5,6}} => {{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => [2,1,4,6,3,5] => 2

{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5},{4,6}} => [1,2,5,6,3,4] => [1,2,4,6,3,5] => 2

{{1,2,5,6},{3,4}} => {{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [3,6,1,2,5,4] => 3

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 7\ q + 8\ q^{2}$

$F_{5} = 12\ q + 35\ q^{2} + 5\ q^{3}$

$F_{6} = 20\ q + 122\ q^{2} + 61\ q^{3}$

Description

The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is St000353The number of inner valleys of a permutation..

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

Wachs-White

Description

A transformation of set partitions due to Wachs and White.
Return the set partition of

$\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection

$\gamma$ .
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with

$(i,j)$ to the cell in the

$i$ -th column from the right and the

$j$ -th row from the top.

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.