Identifier
Values
[1] => [1,0] => [1,1,0,0] => [2,3,1] => 2
[1,1] => [1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => 2
[2] => [1,1,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => 2
[1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 2
[1,2] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 2
[2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 2
[3] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 2
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 3
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 2
[4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 2
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [2,6,7,1,3,4,8,5] => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => 2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [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] => 2
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 2
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 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
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.