Identifier
Values
[.,.] => [1,0] => ([],1) => 1
[.,[.,.]] => [1,0,1,0] => ([(0,1)],2) => 1
[[.,.],.] => [1,1,0,0] => ([(0,1)],2) => 1
[.,[.,[.,.]]] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 1
[.,[[.,.],.]] => [1,0,1,1,0,0] => ([(0,2),(2,1)],3) => 1
[[.,.],[.,.]] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => 1
[[.,[.,.]],.] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => 1
[[[.,.],.],.] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[.,[[[.,.],[.,.]],.]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[.,[[.,[[.,.],[.,.]]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [1,1,0,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[.,[.,[[.,.],.]]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[.,[[.,.],[.,.]]]] => [1,1,0,0,1,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[.,[[.,[.,.]],.]]] => [1,1,0,0,1,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[[.,.],[.,[.,.]]]] => [1,1,0,0,1,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[[.,.],[[.,.],.]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[[.,[.,.]],[.,.]]] => [1,1,0,0,1,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[[.,.],[[.,[.,[.,.]]],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,.],[[[.,.],[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [1,1,0,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,.]],[.,[[.,.],.]]] => [1,1,0,1,0,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,1,0,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,1,0,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [1,1,0,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,0,1,0,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,0,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,[[.,.],.]]],[.,.]] => [1,1,0,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [1,1,0,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[.,[.,[[.,.],[.,.]]]],.] => [1,1,0,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 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 length of the shortest maximal antichain in a poset.
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
parallelogram poset
Description
The cell poset of the parallelogram polyomino corresponding to the Dyck path.
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 cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.