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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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 $\chi(\pi)$.