Identifier
Values
[1,0] => [1,0] => [1,0] => 10 => 1
[1,0,1,0] => [1,1,0,0] => [1,0,1,0] => 1010 => 2
[1,1,0,0] => [1,0,1,0] => [1,1,0,0] => 1100 => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 111000 => 1
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 2
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 101010 => 3
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 110100 => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 1
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 2
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2
[1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0] => 11110000 => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 2
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 4
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 3
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1
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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
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
reflect parallelogram polyomino
Description
Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
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.