- FindStat

Identifier

Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations

Images

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

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

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

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

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

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

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

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

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

[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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5979 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.
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 $\chi(\pi)$.

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

Foata-Han

Description

The Foata-Han bijection for signed permutations.
This map sends the flag major index St001433The flag major index of a signed permutation. to the flag inversion number St001428The number of B-inversions of a signed permutation..