- FindStat

Identifier

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001640: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 11 + 9\ q + 3\ q^{2} + q^{3}$

$F_{5} = 53 + 44\ q + 18\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 309 + 265\ q + 110\ q^{2} + 30\ q^{3} + 5\ q^{4} + q^{5}$

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.

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.