- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000006: 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] => [1,0,1,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,4] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 5

[1,1,4,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 5

[1,4,1,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 5

[4,1,1,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,2,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[1,1,4,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[1,2,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[1,2,4,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[1,4,1,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[1,4,2,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[2,1,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[2,1,4,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[2,4,1,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[4,1,1,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[4,1,2,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

[4,2,1,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 3

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

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

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

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

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

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

[1,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,1,4,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,3,1,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,3,4,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,4,1,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,4,3,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[3,1,1,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[3,1,4,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[3,4,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[4,1,1,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[4,1,3,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[4,3,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see St000012The area of a Dyck path.).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.