- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤ

Values

[1,0] => [1,0] => 10 => [1,2] => 2
[1,0,1,0] => [1,1,0,0] => 1100 => [1,1,3] => 3
[1,1,0,0] => [1,0,1,0] => 1010 => [1,2,2] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 4
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 3
[1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => [1,1,3,2] => 3
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 4
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => 4
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => 4
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => [1,1,2,3,2] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => [1,1,3,1,3] => 3
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 4
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 3
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => [1,1,3,2,2] => 3
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => [1,2,1,3,2] => 3
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 2
[] => [] =>  => [1] => 1

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$F_{1} = q^{2}$

$F_{2} = q^{2} + q^{3}$

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

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

Description

The largest part of an integer composition.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

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

$1 \tilde D_2 0 \tilde D_1$ .