- FindStat

Identifier

Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤ

Values

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

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

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

[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] => 0

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

[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] => 0

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

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

[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] => 2

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

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

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

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

[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] => 1

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

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

[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] => 0

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

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

[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] => 4

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

[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] => 1

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

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

[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] => 0

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

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

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

[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] => 2

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

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

[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] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

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

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

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

[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] => 4

[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] => 5

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

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$F_{2} = 2$

$F_{3} = 4 + q$

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

$F_{5} = 11 + 8\ q + 12\ q^{2} + 8\ q^{3} + 3\ q^{4}$

$F_{6} = 16 + 13\ q + 22\ q^{2} + 27\ q^{3} + 28\ q^{4} + 15\ q^{5} + 10\ q^{6} + q^{7}$

$F_{7} = 22 + 19\ q + 34\ q^{2} + 45\ q^{3} + 66\ q^{4} + 66\ q^{5} + 70\ q^{6} + 52\ q^{7} + 36\ q^{8} + 15\ q^{9} + 4\ q^{10}$

Description

The bounce deficit of a Dyck path.
For a Dyck path

$D$ of semilength

$n$ , this is defined as

$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$
The zeta map Mp00032inverse zeta map sends this statistic to the dinv deficit St000369The dinv deficit of a Dyck path., both are thus equidistributed.

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
$NENNENNNENEEENEE.$