Identifier
-
Mp00233:
Dyck paths
—skew partition⟶
Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000805: Integer compositions ⟶ ℤ
Values
[1,0] => [[1],[]] => [1] => [1] => 1
[1,0,1,0] => [[1,1],[]] => [1,1] => [1,1] => 1
[1,1,0,0] => [[2],[]] => [2] => [2] => 1
[1,0,1,0,1,0] => [[1,1,1],[]] => [1,1,1] => [1,1,1] => 1
[1,0,1,1,0,0] => [[2,1],[]] => [2,1] => [1,2] => 1
[1,1,0,0,1,0] => [[2,2],[1]] => [1,2] => [2,1] => 1
[1,1,0,1,0,0] => [[3],[]] => [3] => [3] => 1
[1,1,1,0,0,0] => [[2,2],[]] => [2,2] => [2,2] => 1
[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => [1,1,1,1] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => [2,1,1] => [1,1,2] => 1
[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1,2,1] => [2,1,1] => 1
[1,0,1,1,0,1,0,0] => [[3,1],[]] => [3,1] => [1,3] => 1
[1,0,1,1,1,0,0,0] => [[2,2,1],[]] => [2,2,1] => [2,1,2] => 2
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1,2] => [1,2,1] => 1
[1,1,0,0,1,1,0,0] => [[3,2],[1]] => [2,2] => [2,2] => 1
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [1,3] => [3,1] => 1
[1,1,0,1,0,1,0,0] => [[4],[]] => [4] => [4] => 1
[1,1,0,1,1,0,0,0] => [[3,3],[1]] => [2,3] => [3,2] => 1
[1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => [1,2,2] => [2,2,1] => 1
[1,1,1,0,0,1,0,0] => [[3,2],[]] => [3,2] => [2,3] => 1
[1,1,1,0,1,0,0,0] => [[2,2,2],[]] => [2,2,2] => [2,2,2] => 1
[1,1,1,1,0,0,0,0] => [[3,3],[]] => [3,3] => [3,3] => 1
[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => [2,1,1,1] => [1,1,1,2] => 1
[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1,2,1,1] => [2,1,1,1] => 1
[1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => [3,1,1] => [1,1,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]] => [2,2,1,1] => [2,1,1,2] => 2
[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => [2,2,1] => [2,1,2] => 2
[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [1,3,1] => [3,1,1] => 1
[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => [4,1] => [1,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1,2,2,1] => [2,2,1,1] => 1
[1,0,1,1,1,0,0,1,0,0] => [[3,2,1],[]] => [3,2,1] => [2,1,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1],[]] => [2,2,2,1] => [2,2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]] => [3,3,1] => [3,1,3] => 2
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1,2] => [1,1,2,1] => 1
[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => [2,1,2] => [1,2,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [1,2,2] => [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => [3,2] => [2,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [2,2,2] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [1,1,3] => [1,3,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2,3] => [3,2] => 1
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [1,4] => [4,1] => 1
[1,1,0,1,0,1,0,1,0,0] => [[5],[]] => [5] => [5] => 1
[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2,4] => [4,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [1,2,3] => [2,3,1] => 1
[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [3,3] => [3,3] => 1
[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => [2,2,3] => [2,3,2] => 1
[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => [3,4] => [4,3] => 1
[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1,2,2] => [1,2,2,1] => 1
[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [2,2,2] => [2,2,2] => 1
[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [1,3,2] => [3,2,1] => 1
[1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => [4,2] => [2,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => [2,3,2] => [3,2,2] => 1
[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]] => [1,2,2,2] => [2,2,2,1] => 1
[1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => [3,2,2] => [2,2,3] => 1
[1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2],[]] => [2,2,2,2] => [2,2,2,2] => 1
[1,1,1,0,1,1,0,0,0,0] => [[3,3,2],[]] => [3,3,2] => [3,2,3] => 2
[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [1,3,3] => [3,3,1] => 1
[1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => [4,3] => [3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]] => [2,3,3] => [3,3,2] => 1
[1,1,1,1,0,1,0,0,0,0] => [[4,4],[]] => [4,4] => [4,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [[3,3,3],[]] => [3,3,3] => [3,3,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => [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]] => [1,2,1,1,1] => [2,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]] => [3,1,1,1] => [1,1,1,3] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]] => [2,2,1,1,1] => [2,1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1,2,1,1] => [1,2,1,1,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => [2,2,1,1] => [2,1,1,2] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [1,3,1,1] => [3,1,1,1] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]] => [4,1,1] => [1,1,4] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [[3,3,1,1],[1]] => [2,3,1,1] => [3,1,1,2] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => [1,2,2,1,1] => [2,2,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1],[]] => [3,2,1,1] => [2,1,1,3] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1],[]] => [2,2,2,1,1] => [2,2,1,1,2] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1],[]] => [3,3,1,1] => [3,1,1,3] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1,2,1] => [1,1,2,1,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => [2,1,2,1] => [1,2,1,2] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [1,2,2,1] => [2,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]] => [3,2,1] => [2,1,3] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => [2,2,2,1] => [2,2,1,2] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [1,1,3,1] => [1,3,1,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [1,4,1] => [4,1,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => [5,1] => [1,5] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2,4,1] => [4,1,2] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [1,2,3,1] => [2,3,1,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]] => [3,3,1] => [3,1,3] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]] => [2,2,3,1] => [2,3,1,2] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]] => [3,4,1] => [4,1,3] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => [1,1,2,2,1] => [1,2,2,1,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => [2,2,2,1] => [2,2,1,2] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]] => [1,3,2,1] => [3,2,1,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1],[]] => [4,2,1] => [2,1,4] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]] => [2,3,2,1] => [3,2,1,2] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]] => [1,2,2,2,1] => [2,2,2,1,1] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1],[]] => [3,2,2,1] => [2,2,1,3] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1],[]] => [2,2,2,2,1] => [2,2,2,1,2] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1],[]] => [3,3,2,1] => [3,2,1,3] => 2
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Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Map
rotate front to back
Description
The front to back rotation of the entries of an integer composition.
Map
row lengths
Description
The sequence of row lengths from top to bottom.
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 $\gamma(D)$.
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 $\gamma(D)$.
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