Identifier
Values
[1,2] => [1,0,1,0] => ([(0,1)],2) => 1
[2,1] => [1,1,0,0] => ([(0,1)],2) => 1
[1,2,3] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 2
[1,3,2] => [1,0,1,1,0,0] => ([(0,2),(2,1)],3) => 2
[2,1,3] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => 2
[2,3,1] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => 2
[3,1,2] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[3,2,1] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[1,2,3,4] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[1,4,3,2] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[2,1,3,4] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 3
[2,3,1,4] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[2,4,3,1] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[3,1,2,4] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[3,1,4,2] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[3,2,1,4] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[3,2,4,1] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
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Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
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.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.