Identifier
-
Mp00233:
Dyck paths
—skew partition⟶
Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001236: Integer compositions ⟶ ℤ
Values
[1,0] => [[1],[]] => [1] => [1] => 1
[1,0,1,0] => [[1,1],[]] => [1,1] => [2] => 1
[1,1,0,0] => [[2],[]] => [2] => [1,1] => 2
[1,0,1,0,1,0] => [[1,1,1],[]] => [1,1,1] => [3] => 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] => [1,1,1] => 3
[1,1,1,0,0,0] => [[2,2],[]] => [2,2] => [1,2,1] => 2
[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => [1,1,1,1] => [4] => 1
[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => [2,1,1] => [1,3] => 1
[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1,2,1] => [2,2] => 1
[1,0,1,1,0,1,0,0] => [[3,1],[]] => [3,1] => [1,1,2] => 1
[1,0,1,1,1,0,0,0] => [[2,2,1],[]] => [2,2,1] => [1,2,2] => 1
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1,2] => [3,1] => 1
[1,1,0,0,1,1,0,0] => [[3,2],[1]] => [2,2] => [1,2,1] => 2
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [1,3] => [2,1,1] => 1
[1,1,0,1,0,1,0,0] => [[4],[]] => [4] => [1,1,1,1] => 4
[1,1,0,1,1,0,0,0] => [[3,3],[1]] => [2,3] => [1,2,1,1] => 2
[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] => [1,1,2,1] => 2
[1,1,1,0,1,0,0,0] => [[2,2,2],[]] => [2,2,2] => [1,2,2,1] => 2
[1,1,1,1,0,0,0,0] => [[3,3],[]] => [3,3] => [1,1,2,1,1] => 3
[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [5] => 1
[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => [2,1,1,1] => [1,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1,2,1,1] => [2,3] => 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] => [1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1,2,1] => [3,2] => 1
[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => [2,2,1] => [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [1,3,1] => [2,1,2] => 1
[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => [4,1] => [1,1,1,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]] => [2,3,1] => [1,2,1,2] => 1
[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1,2,2,1] => [2,2,2] => 1
[1,0,1,1,1,0,0,1,0,0] => [[3,2,1],[]] => [3,2,1] => [1,1,2,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1,2] => [4,1] => 1
[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => [2,1,2] => [1,3,1] => 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] => [1,1,2,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [2,2,2] => [1,2,2,1] => 2
[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [1,1,3] => [3,1,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2,3] => [1,2,1,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [1,4] => [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0] => [[5],[]] => [5] => [1,1,1,1,1] => 5
[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2,4] => [1,2,1,1,1] => 2
[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [1,2,3] => [2,2,1,1] => 1
[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [3,3] => [1,1,2,1,1] => 3
[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1,2,2] => [3,2,1] => 1
[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [2,2,2] => [1,2,2,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [1,3,2] => [2,1,2,1] => 1
[1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => [4,2] => [1,1,1,2,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [1,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => [1,2,1,1,1] => [2,4] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]] => [3,1,1,1] => [1,1,4] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1,2,1,1] => [3,3] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => [2,2,1,1] => [1,2,3] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [1,3,1,1] => [2,1,3] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]] => [4,1,1] => [1,1,1,3] => 1
[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] => [4,2] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => [2,1,2,1] => [1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [1,2,2,1] => [2,2,2] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]] => [3,2,1] => [1,1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [1,1,3,1] => [3,1,2] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2,3,1] => [1,2,1,2] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [1,4,1] => [2,1,1,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => [5,1] => [1,1,1,1,2] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1,2] => [5,1] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [2,1,1,2] => [1,4,1] => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [1,2,1,2] => [2,3,1] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2],[1,1]] => [3,1,2] => [1,1,3,1] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [1,1,2,2] => [3,2,1] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,2,2] => [1,2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [1,3,2] => [2,1,2,1] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2],[1]] => [4,2] => [1,1,1,2,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [1,1,1,3] => [4,1,1] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3],[2,2]] => [2,1,3] => [1,3,1,1] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [1,2,3] => [2,2,1,1] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3],[2]] => [3,3] => [1,1,2,1,1] => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [1,1,4] => [3,1,1,1] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [2,4] => [1,2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [1,5] => [2,1,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => [6] => [1,1,1,1,1,1] => 6
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Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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 $\gamma(D)$.
Map
row lengths
Description
The sequence of row lengths from top to bottom.
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