- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000884: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 2\ q^{2}$

$F_{5} = 1 + 26\ q + 15\ q^{2}$

$F_{6} = 1 + 57\ q + 69\ q^{2} + 5\ q^{3}$

Description

The number of isolated descents of a permutation.
A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.