Identifier
Values
[(1,2)] => [2,1] => [1,2] => [1,2] => 2
[(1,2),(3,4)] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[(1,3),(2,4)] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[(1,4),(2,3)] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [5,6,2,1,4,3] => [2,6,4,1,5,3] => 3
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [5,6,1,2,4,3] => [6,1,4,2,5,3] => 3
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [4,1,6,2,5,3] => [4,6,5,1,2,3] => 2
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [1,4,6,2,5,3] => [1,6,5,4,2,3] => 2
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 2
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [3,1,2,6,5,4] => [3,1,2,5,6,4] => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 2
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [4,3,6,5,1,2] => [5,1,4,3,6,2] => 2
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [3,4,6,5,1,2] => [5,1,3,4,6,2] => 2
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [2,4,6,1,5,3] => [6,2,5,4,1,3] => 2
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [2,3,1,6,5,4] => [3,2,1,5,6,4] => 2
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 2
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 2
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [2,3,4,1,6,7,8,5] => 2
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 breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
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 $\chi(\pi)$.
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.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.