Identifier
Values
[1] => [1,0] => [[1],[]] => 1
[1,1] => [1,0,1,0] => [[1,1],[]] => 1
[2] => [1,1,0,0] => [[2],[]] => 1
[1,1,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[1,2] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[2,1] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[3] => [1,1,1,0,0,0] => [[2,2],[]] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
[3,1] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[4] => [1,1,1,1,0,0,0,0] => [[3,3],[]] => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1,1]] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1]] => 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1],[]] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1],[]] => 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1,1],[1]] => 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1,1],[1,1]] => 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1,1],[1]] => 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1,1],[1,1,1]] => 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1,1],[1,1]] => 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1],[2,1]] => 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,1],[1,1,1,1]] => 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,1],[1,1,1]] => 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,1],[2,1,1]] => 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,1],[2,2,1]] => 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1],[2,1]] => 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1,1,1,1]] => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2],[1,1,1,1]] => 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2],[2,1,1,1]] => 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2],[2,2,1,1]] => 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2],[2,1,1]] => 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2],[2,2,2,1]] => 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2],[2,2,1]] => 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2],[3,2,1]] => 1
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Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
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
bounce path
Description
The bounce path determined by an integer composition.