Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => [[2],[]] => 2
[2] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => [[2,2],[]] => 2
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [[3],[]] => 2
[2,1] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => [[2,2],[1]] => 3
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 3
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 2
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 3
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 3
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 3
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 3
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 4
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 3
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 4
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 4
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 4
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 5
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
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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 $\gamma(D)$.