Identifier
Values
[1,0] => [1,0] => [1] => ([],1) => 1
[1,0,1,0] => [1,1,0,0] => [1,2] => ([(0,1)],2) => 1
[1,1,0,0] => [1,0,1,0] => [2,1] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[1,1,1,0,0,0] => [1,1,0,1,0,0] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],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 number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
Delest-Viennot-inverse
Description
Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
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.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\beta^{(-1)}\circ\gamma)(D)$.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.