- FindStat

Identifier

Mp00286: Ordered set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => [1,2]

[{1,2}] => [1,2] => [1,2] => [1,2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5316 entries. <<<

Map

to permutation

Description

Return the permutation obtained by concatenating the blocks sorted into increasing order.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

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 $\chi(\pi)$.