- FindStat

Identifier

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤ

Values

[1,0] => [1,0] => 10 => 01 => 1
[1,0,1,0] => [1,1,0,0] => 1100 => 1011 => 2
[1,1,0,0] => [1,0,1,0] => 1010 => 1001 => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 010111 => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 011011 => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 010001 => 3
[1,1,0,1,0,0] => [1,0,1,0,1,0] => 101010 => 011001 => 2
[1,1,1,0,0,0] => [1,1,0,1,0,0] => 110100 => 010011 => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 10101111 => 4
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 10010111 => 3
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10111011 => 3
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10011011 => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 10110111 => 3
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 10100001 => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 10010001 => 3
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 10111001 => 3
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 10011001 => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 10110001 => 3
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10100011 => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10010011 => 2
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 10100111 => 3
[1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10110011 => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0101011111 => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0100010111 => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0100101111 => 4
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0100011011 => 3
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0101000001 => 5
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0110100001 => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0110011001 => 2
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0101110001 => 3
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0110110001 => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0101000111 => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0100100111 => 3
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0100110011 => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 101110100001 => 4
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 101110010001 => 3
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 100110011001 => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 101100011001 => 3
[1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 100101110001 => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 100110110001 => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 101011100001 => 4
[1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 101101100001 => 4
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 101001110001 => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 101100110001 => 3

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$F_{1} = q$

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

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

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

Description

The length of the longest constant subword.

Map

zeros to flag zeros

Description

Return a binary word of the same length, such that the number of occurrences of

$10$ in the word obtained by prepending the reverse of the complement equals the number of

$0$ s in the original word.
For example, the image of the word

$w=1\dots1$ is

$1\dots1$ , because

$w$ has no zeros, and

$1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor

$10$ .
On the other hand,

$0\dots0$ must be mapped to

$10\dots10$ if the length is even, and

$010\dots10$ if it is odd.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

Delest-Viennot-inverse

Description

Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\beta^{(-1)}\circ\gamma)(D)$ .