- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000842: Permutations ⟶ ℤ

Values

[1,0,1,0] => [1,2] => 2

[1,1,0,0] => [2,1] => 2

[1,0,1,0,1,0] => [1,2,3] => 2

[1,0,1,1,0,0] => [1,3,2] => 2

[1,1,0,0,1,0] => [2,1,3] => 2

[1,1,0,1,0,0] => [2,3,1] => 2

[1,1,1,0,0,0] => [3,2,1] => 2

[1,0,1,0,1,0,1,0] => [1,2,3,4] => 2

[1,0,1,0,1,1,0,0] => [1,2,4,3] => 2

[1,0,1,1,0,0,1,0] => [1,3,2,4] => 2

[1,0,1,1,0,1,0,0] => [1,3,4,2] => 2

[1,0,1,1,1,0,0,0] => [1,4,3,2] => 2

[1,1,0,0,1,0,1,0] => [2,1,3,4] => 2

[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2

[1,1,0,1,0,0,1,0] => [2,3,1,4] => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => 2

[1,1,0,1,1,0,0,0] => [2,4,3,1] => 2

[1,1,1,0,0,0,1,0] => [3,2,1,4] => 2

[1,1,1,0,0,1,0,0] => [3,2,4,1] => 2

[1,1,1,0,1,0,0,0] => [3,4,2,1] => 2

[1,1,1,1,0,0,0,0] => [4,3,2,1] => 2

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 2

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 2

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 2

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 2

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 2

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 2

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 2

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 2

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 2

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 2

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 2

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 2

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 2

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 2

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 2

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 2

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 2

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 2

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 2

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 2

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 2

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 2

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 2

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 2

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 2

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 2

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 2

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 2

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 2

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 2

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 2

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 2

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 2

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 2

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 2

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 2

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 2

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 2

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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

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).