Identifier
Values
[1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1,0,0] => [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,1,2,4] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,4,2,1,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,4,5,2,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,1,5,3,2] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,2,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,5,3,2,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,6,1,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,5,1,6,2,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,4,2,1,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,5,2,3,1,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,4,6,1,2,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,6,1,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,5,1,2,6,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,5,2,1,4,3] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,6,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,2,5,3,1,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,6,1,5,3] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,5,6,2,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,6,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,2,3,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,1,5,6,3,2] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,4,6,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,6,5,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,1,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,4,6,2,5,1] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,3,6,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,5,6,3,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,1,6,2,3,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,2,5,6,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,5,3,1,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,1,6,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,2,1,6,4,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,1,6,3,4,2] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [7,4,2,1,3,5,6] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => 4
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => [5,1,2,8,3,4,6,7] => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [8,1,2,5,3,4,6,7] => [8,1,5,3,2,4,6,7] => 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [8,1,2,5,6,3,4,7] => [8,1,6,3,2,5,4,7] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => [3,5,1,7,2,8,4,6] => 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [8,1,4,2,3,7,5,6] => [8,4,2,1,7,5,3,6] => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [8,1,5,2,7,3,4,6] => [8,5,7,3,1,4,2,6] => 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [8,1,4,6,2,7,3,5] => [8,6,7,1,3,4,2,5] => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [8,1,4,5,6,7,2,3] => [8,7,2,1,4,5,6,3] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [8,2,1,3,4,5,6,7] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,6] => 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,3,8,1,4,5,6,7] => [8,2,3,1,4,5,6,7] => 3
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,3,5,1,6,7,8,4] => [5,2,3,8,1,6,7,4] => 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [2,7,4,1,6,3,8,5] => [4,2,7,6,8,3,1,5] => 3
[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [6,5,4,1,2,8,3,7] => [4,1,5,6,8,2,3,7] => 2
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [8,3,5,1,6,2,4,7] => [5,6,8,2,3,1,4,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => [9,2,1,3,4,5,6,7,8] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,9,1,4,5,6,7,8] => [9,2,3,1,4,5,6,7,8] => 3
[] => [1] => [1] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,10,8] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => [10,2,1,3,4,5,6,7,8,9] => 2
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Description
The number of cycles in the cycle decomposition of a permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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 $\chi(\pi)$.