Identifier
-
Mp00233:
Dyck paths
—skew partition⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤ (values match St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.)
Values
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [[3,3,3,1],[2]] => [2] => [1,0,1,0] => 2
[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] => [1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [[4,4,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 2
[1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [[5,5],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [[4,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,1,1,0,0,1,0,0,1,0] => [[4,4,3],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,1,1,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,1,1,1,0,0,0,0,1,0] => [[4,4,4],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [[4,3,2],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [[4,4,2],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [[4,4,2],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,0,0,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,0,1,0,0,1,0,0,1,0] => [[3,3,2,2],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,0,1,1,0,0,0,0,1,0] => [[3,3,3,2],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [[4,3,3],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [[4,4,3],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [[4,4,3],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [[4,4,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4],[2]] => [2] => [1,0,1,0] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1,1],[2]] => [2] => [1,0,1,0] => 2
[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] => [1,1,0,1,0,0] => 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2,1],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3,1],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1],[2]] => [2] => [1,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4,1],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5,1],[4]] => [4] => [1,0,1,0,1,0,1,0] => 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,1],[2]] => [2] => [1,0,1,0] => 2
[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]] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[5,5,2],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[6,3],[2]] => [2] => [1,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2
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search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
inner shape
Description
The inner shape of a 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 $\gamma(D)$.
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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