- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000237: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,5,1,4,2] => [4,2,5,1,3] => [5,4,1,2,3] => [4,5,2,1,3] => 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 small exceedances.
This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Map

Inverse Kreweras complement

Description

Sends the permutation

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

$c\pi^{-1}$ where

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

Map

descent views to invisible inversion bottoms

Description

Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation

$\sigma$ is a pair

$i < j$ such that

$i < \sigma(j) < \sigma(i)$ . The element

$\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation

$\pi$ is an element

$\pi(j)$ such that

$\pi(i+1) < \pi(j) < \pi(i)$ , and additionally the smallest element in the decreasing run containing

$\pi(i)$ is smaller than the smallest element in the decreasing run containing

$\pi(j)$ .
This map is a bijection

$\chi:\mathfrak S_n \to \mathfrak S_n$ , such that

the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$ ,
the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$ ,
the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$ ,
the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$ .

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.