Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤ
Values
[1,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[2,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[1,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[3,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[2,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[2,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[1,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[4,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[3,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[3,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[2,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[2,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[5,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[4,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[4,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[3,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[3,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[3,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[2,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[2,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[6,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[5,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[5,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[4,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[4,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[4,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[3,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[3,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[3,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[2,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[7,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[6,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[6,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[5,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[5,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[5,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[4,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[4,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[3,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[3,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 2
[3,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[8,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[7,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[7,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[6,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[6,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[6,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[5,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[5,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[5,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[4,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 2
[4,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[3,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[9,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[8,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[8,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[7,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[7,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[7,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[6,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[6,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[6,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[5,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[5,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 2
[5,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[4,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[3,3,3,1] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 2
[3,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 2
[3,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[10,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[9,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[9,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[8,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[8,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[8,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[7,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[7,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[7,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[6,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[6,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 2
[6,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[5,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[4,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 2
[4,3,3,1] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 2
[4,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 2
[4,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[3,3,3,2] => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 2
[3,3,3,1,1] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 2
[3,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[11,1] => [1] => [1,0,1,0] => [[1,1],[]] => 1
[10,2] => [2] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[10,1,1] => [1,1] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[9,3] => [3] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[9,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[9,1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[8,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[8,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 2
[8,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[7,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
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searching the database for the individual values of this statistic
Description
The number of inner corners of a skew partition.
Map
first row removal
Description
Removes the first entry of an integer 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 $\gamma(D)$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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