- FindStat

Identifier

Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤ (values match St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 10

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

[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 7

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

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

[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 8

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

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

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

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 9

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

[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 7

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

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

[1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 8

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

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 10

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 11

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 7

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 12

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 8

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 9

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 7

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

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

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 13

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 9

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 10

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 7

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 11

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 8

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 9

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 7

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

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

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

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

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

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 8

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

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 7

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

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

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

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

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

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

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Description

The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

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
$$NENNENNNENEEENEE.$$