- FindStat

Identifier

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on signed permutations.
This map is a surjection onto the set of involutions, i.e., the set of signed permutations

$\pi$ for which

$\pi = \pi^{-1}$ .

Map

permutation

Description

The permutation obtained by forgetting the colours.

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.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
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 maximima 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 deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$ .