Identifier
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [2,3,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 3
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 3
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 3
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 4
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 4
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 3
[3,4,2,1] => [2,4,3,1] => [2,4,3,1] => 3
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 4
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 4
[4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 3
[4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 3
[1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 3
[1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 4
[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 3
[1,4,5,3,2] => [1,3,5,4,2] => [1,3,5,4,2] => 3
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 3
[1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 4
[1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 3
[1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 4
[1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 3
[1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
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Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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 $\chi(\pi)$.