Identifier
Values
[1,0] => [1,0] => [1,1,0,0] => [[2],[]] => 0
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => [[3],[]] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [[2,2,2],[]] => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 0
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [[3,3],[]] => 2
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => 1
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => 1
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => 2
[1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2],[]] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => 1
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]] => 0
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Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
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.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
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.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.