Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001398: Posets ⟶ ℤ
Values
[1] => [1,0] => [1,0] => ([],1) => 0
[2] => [1,0,1,0] => [1,1,0,0] => ([(0,1)],2) => 0
[1,1] => [1,1,0,0] => [1,0,1,0] => ([(0,1)],2) => 0
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => 0
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 0
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 4
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 0
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 4
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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.
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
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
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