Identifier
Values
[1,0] => [(1,2)] => [2,1] => [2,1] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => [2,4,3,1] => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,4,6,5,3] => 3
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [2,4,3,1,6,5] => 3
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [4,6,3,5,1,2] => 3
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [2,3,6,4,5,1] => 4
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 4
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 4
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [2,1,4,5,8,6,7,3] => 5
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 4
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [2,4,3,1,6,8,7,5] => 4
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [2,3,6,4,5,1,8,7] => 5
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => 4
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
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
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 $\chi(\pi)$.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.