Identifier
Values
{{1,2}} => [2,1] => [2,1] => [1,2] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => [1,3,2] => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => [1,2,3] => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => [1,2,3] => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,2,3] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 2
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
{{1,2,4},{3}} => [2,4,3,1] => [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [4,3,2,1] => [1,4,2,3] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => [1,2,3,4] => 0
{{1,4},{2,3}} => [4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [3,4,2,1] => [1,3,2,4] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => [1,2,3,4] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
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Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
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 $\chi(\pi)$.