Identifier
Values
[1,0] => [1,0] => ([],1) => 1
[1,0,1,0] => [1,1,0,0] => ([(0,1)],2) => 1
[1,1,0,0] => [1,0,1,0] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => 1
[1,1,0,0,1,0] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => ([(0,2),(2,1)],3) => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 1
[1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[1,1,0,1,0,0,1,0] => [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] => [1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 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 the Hasse diagram for the poset.
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
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.