- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000039: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 4\ q + 2\ q^{2}$

$F_{5} = 16 + 12\ q + 9\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 32 + 32\ q + 30\ q^{2} + 20\ q^{3} + 12\ q^{4} + 4\ q^{5} + 2\ q^{6}$

Description

The number of crossings of a permutation.
A crossing of a permutation

$\pi$ is given by a pair

$(i,j)$ such that either

$i < j \leq \pi(i) \leq \pi(j)$ or

$\pi(i) < \pi(j) < i < j$ .
Pictorially, the diagram of a permutation is obtained by writing the numbers from

$1$ to

$n$ in this order on a line, and connecting

$i$ and

$\pi(i)$ with an arc above the line if

$i\leq\pi(i)$ and with an arc below the line if

$i > \pi(i)$ . Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.