Identifier
Values
[.,.] => [1,0] => [[1],[]] => 1
[.,[.,.]] => [1,1,0,0] => [[2],[]] => 1
[[.,.],.] => [1,0,1,0] => [[1,1],[]] => 1
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => [[3],[]] => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[[.,[.,.]],.] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[[[.,.],.],.] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [[4],[]] => 1
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 1
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 1
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 1
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 1
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 1
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[[[[[.,.],.],.],.],.] => [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 connected components of a skew partition.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
  • a leaf is associated with an empty Dyck path,
  • a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
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 $\gamma(D)$.