- FindStat

Identifier

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000308: Permutations ⟶ ℤ

Values

[1,0] => [1,0] => [2,1] => [1,2] => 2
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [1,2,3] => 3
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 4
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => 3
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [1,4,2,3] => 3
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => [1,2,4,3] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => 4
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,4,5,2,3] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => 4
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,5,3,2] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,3,4] => 4
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,5,3,2,4] => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,2,5,3,4] => 4
[1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,2,3,5,4] => 4
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,5,4,2] => 3
[1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,3,4,2] => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,4,2,3] => 3
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,2,5,4,3] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,4,5,6,2,3] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,2,4,5,6,3] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 4
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,5,6,2,3,4] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,2,5,6,3,4] => 4
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,2,3,5,6,4] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,5,6,4,2] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,6,3,4,2] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,5,6,4,2,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,4,5] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,2,6,4,3,5] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,4,2,3,5] => 4
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,2,6,3,4,5] => 5
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,6,4,2,3,5] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,2,3,6,4,5] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 4
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,3,6,4,5,2] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,4,6,5,2,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,2,4,6,5,3] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,6,5,3,2] => 3
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,3,4,5,2] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,6,5,2,3,4] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,2,6,4,5,3] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,2,6,5,3,4] => 4
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,6,5,4,2] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,5,3,4,2] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 7
[] => [] => [1] => [1] => 1

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$F_{1} = q^{2}$

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

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

$F_{4} = q^{2} + 7\ q^{3} + 5\ q^{4} + q^{5}$

$F_{5} = q^{2} + 15\ q^{3} + 18\ q^{4} + 7\ q^{5} + q^{6}$

Description

The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

zeta map

Description

The zeta map on Dyck paths.
The zeta map

$\zeta$ is a bijection on Dyck paths of semilength

$n$ .
It was defined in [1, Theorem 1], see also [2, Theorem 3.15] and sends the bistatistic (area, dinv) to the bistatistic (bounce, area). It is defined by sending a Dyck path

$D$ with corresponding area sequence

$a=(a_1,\ldots,a_n)$ to a Dyck path as follows:

First, build an intermediate Dyck path consisting of $d_1$ north steps, followed by $d_1$ east steps, followed by $d_2$ north steps and $d_2$ east steps, and so on, where $d_i$ is the number of $i-1$ 's within the sequence $a$ .
For example, given $a=(0,1,2,2,2,3,1,2)$ , we build the path
$NE\ NNEE\ NNNNEEEE\ NE.$
Next, the rectangles between two consecutive peaks are filled. Observe that such the rectangle between the $k$ th and the $(k+1)$ st peak must be filled by $d_k$ east steps and $d_{k+1}$ north steps. In the above example, the rectangle between the second and the third peak must be filled by $2$ east and $4$ north steps, the $2$ being the number of $1$ 's in $a$ , and $4$ being the number of $2$ 's. To fill such a rectangle, scan through the sequence a from left to right, and add east or north steps whenever you see a $k-1$ or $k$ , respectively. So to fill the $2\times 4$ rectangle, we look for $1$ 's and $2$ 's in the sequence and see $122212$ , so this rectangle gets filled with $ENNNEN$ .
The complete path we obtain in thus
$NENNENNNENEEENEE.$