Identifier
Values
[.,.] => [.,.] => [1,0] => [[1],[]] => 1
[.,[.,.]] => [[.,.],.] => [1,1,0,0] => [[2],[]] => 1
[[.,.],.] => [.,[.,.]] => [1,0,1,0] => [[1,1],[]] => 1
[.,[.,[.,.]]] => [[.,[.,.]],.] => [1,1,0,1,0,0] => [[3],[]] => 1
[.,[[.,.],.]] => [[[.,.],.],.] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[[.,.],[.,.]] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[[.,[.,.]],.] => [.,[[.,.],.]] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[[[.,.],.],.] => [[.,.],[.,.]] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[.,[.,[.,[.,.]]]] => [[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [[4],[]] => 1
[.,[.,[[.,.],.]]] => [[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[.,[[.,.],[.,.]]] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[[.,.],[.,[.,.]]] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[[.,.],[[.,.],.]] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 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
[[.,[.,[.,.]]],.] => [.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[[.,[[.,.],.]],.] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[[[.,.],[.,.]],.] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
[[[.,[.,.]],.],.] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 1
[[[[.,.],.],.],.] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[.,[.,[.,[.,[.,.]]]]] => [[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1
[[.,.],[.,[.,[.,.]]]] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[[.,.],[.,[[.,.],.]]] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[[.,.],[[.,.],[.,.]]] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[[.,.],[[.,[.,.]],.]] => [.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[[.,[.,.]],[.,[.,.]]] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[[.,[.,.]],[[.,.],.]] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[[[.,.],.],[.,[.,.]]] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[[[.,.],.],[[.,.],.]] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 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
[[[.,[.,.]],.],[.,.]] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 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
[[[.,[.,.]],[.,.]],.] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 1
[[[.,[.,[.,.]]],.],.] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 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 Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ 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 $\gamma(D)$.
Map
right rotate
Description
Return the result of right rotation applied to a binary tree.
Right rotation on binary trees is defined as follows: Let $T$ be a binary tree such that the left child of the root of $T$ is a node. Let $C$ be the right child of the root of $T$, and let $A$ and $B$ be the left and right children of the left child of the root of $T$. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the right rotation of $T$ is the binary tree in which the left child of the root is $A$, whereas the right child of the root is a node whose left and right children are $B$ and $C$.