- FindStat

Identifier

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 6697 entries. <<<

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

to signed permutation

Description

The signed permutation with all signs positive.

Map

inverse Kreweras complement

Description

The inverse Kreweras complement of a signed permutation.
This is the signed permutation $c \pi^{-1}$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.