Identifier
Values
[1] => [1,0] => [1,0] => [[1],[]] => 1
[2] => [1,0,1,0] => [1,1,0,0] => [[2],[]] => 1
[1,1] => [1,1,0,0] => [1,0,1,0] => [[1,1],[]] => 1
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[2,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [[2,2],[1]] => 2
[3,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of inner corners of a skew partition.
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.
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
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.