- 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

>>> Load all 18248 entries. <<<

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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 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

$1 \tilde D_2 0 \tilde D_1$ .