- FindStat

Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001948: Permutations ⟶ ℤ

Values

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

$F_{4} = 2 + 4\ q + 3\ q^{2} + q^{3}$

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

Description

The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation

$\pi$ is a double ascent of the augmented permutation

$\tilde\pi$ obtained from

$\pi$ by adding an initial

$0$ .
A double ascent of

$\tilde\pi$ then is a position

$i$ such that

$\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$ .

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

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

Description

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

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .