- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 6 + 6\ q + 3\ q^{2} + q^{3}$

$F_{4} = 24 + 36\ q + 30\ q^{2} + 20\ q^{3} + 10\ q^{4} + 4\ q^{5} + q^{6}$

$F_{5} = 120 + 240\ q + 270\ q^{2} + 240\ q^{3} + 180\ q^{4} + 120\ q^{5} + 70\ q^{6} + 35\ q^{7} + 15\ q^{8} + 5\ q^{9} + q^{10}$

$F_{6} = 720 + 1800\ q + 2520\ q^{2} + 2730\ q^{3} + 2520\ q^{4} + 2100\ q^{5} + 1610\ q^{6} + 1140\ q^{7} + 750\ q^{8} + 455\ q^{9} + 252\ q^{10} + 126\ q^{11} + 56\ q^{12} + 21\ q^{13} + 6\ q^{14} + q^{15}$

Description

The bounce statistic of a Dyck path.
The bounce path

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ .
The points where

$D'$ touches the

$x$ -axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ .
In particular, the bounce statistics of

$D$ and

$D'$ coincide.

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
$NENNENNNENEEENEE.$

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.