- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤ

Values

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

[1,1] => [1,1,0,0] => [1,0,1,0] => 1010 => 2

[1,2] => [1,0,1,0] => [1,1,0,0] => 1100 => 1

[2,1] => [1,0,1,0] => [1,1,0,0] => 1100 => 1

[1,1,1] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 3

[1,1,2] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 2

[1,2,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 2

[2,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 2

[1,1,3] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => 2

[1,3,1] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => 2

[3,1,1] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => 2

[1,2,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 1

[2,1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 1

[2,2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 1

[1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[1,3,2] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[2,1,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[2,3,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[3,1,2] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[3,2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 1

[1,1,1,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 4

[1,1,1,2] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 3

[1,1,2,1] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 3

[1,2,1,1] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 3

[2,1,1,1] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 3

[1,1,1,3] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 3

[1,1,3,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 3

[1,3,1,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 3

[3,1,1,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 3

[1,1,1,4] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 3

[1,1,4,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 3

[1,4,1,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 3

[4,1,1,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 3

[1,1,2,2] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[1,2,1,2] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[1,2,2,1] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[2,1,1,2] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[2,1,2,1] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[2,2,1,1] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[1,1,2,3] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,1,3,2] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,2,1,3] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,2,3,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,3,1,2] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,3,2,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[2,1,1,3] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[2,1,3,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[2,3,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[3,1,1,2] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[3,1,2,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[3,2,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2

[1,1,2,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,1,4,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,2,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,2,4,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,4,1,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,4,2,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[2,1,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[2,1,4,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[2,4,1,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[4,1,1,2] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[4,1,2,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[4,2,1,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[1,1,3,3] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[1,3,1,3] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[1,3,3,1] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[3,1,1,3] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[3,1,3,1] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[3,3,1,1] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[1,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,1,4,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,3,1,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,3,4,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,4,1,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,4,3,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[3,1,1,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[3,1,4,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[3,4,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[4,1,1,3] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[4,1,3,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[4,3,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2

[1,2,2,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1

[2,1,2,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1

[2,2,1,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1

[2,2,2,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1

[1,2,2,3] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[1,2,3,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[1,3,2,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,1,2,3] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,1,3,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,2,1,3] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,2,3,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,3,1,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[2,3,2,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[3,1,2,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[3,2,1,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[3,2,2,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1

[1,2,2,4] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1

[1,2,4,2] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1

[1,4,2,2] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1

[2,1,2,4] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1

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$F_{1} = q$

$F_{2} = 2\ q + q^{2}$

$F_{3} = 9\ q + 6\ q^{2} + q^{3}$

$F_{4} = 64\ q + 48\ q^{2} + 12\ q^{3} + q^{4}$

Description

Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.

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

$1 \tilde D_2 0 \tilde D_1$ .

Map

to binary word

Description

Return the Dyck word as binary word.