Identifier
Values
[1,0] => [1,1,0,0] => [1,0,1,0] => [[1,1],[]] => 2
[1,0,1,0] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => [[3],[]] => 2
[1,1,0,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,1,1],[]] => 2
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 3
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 3
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 2
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 2
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 4
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 3
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 3
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 3
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 2
[1,1,1,1,0,0,0,0] => [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],[]] => 2
[] => [1,0] => [1,0] => [[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 corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.