- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ 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] => [1,3,2] => [1,3,2] => [1,3,2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7307 entries. <<<

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

Map

inverse

Description

Sends a permutation to its inverse.

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 $\chi(\pi)$.