- 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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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

to Dyck path

Description

Sends a parking function to its support Dyck word.

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
$$NENNENNNENEEENEE.$$