Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤ
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) => 2
[[],[],[],[]] => [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) => 2
[[[]],[],[]] => [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) => 2
[[[],[],[]]] => [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) => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => ([(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] => ([(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) => 2
[[],[[]],[],[]] => [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) => 2
[[],[[],[],[]]] => [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) => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[[[]],[],[],[]] => [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) => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => ([(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) => 2
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[[[],[],[]],[]] => [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) => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[[[],[],[],[]]] => [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) => 2
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[[[[]],[[]]]] => [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) => 2
[[],[],[],[],[],[]] => [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) => 2
[[],[],[[],[],[]]] => [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) => 2
[[],[[]],[],[],[]] => [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) => 2
[[],[[],[],[],[]]] => [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) => 2
[[[]],[],[],[],[]] => [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) => 2
[[[]],[[],[],[]]] => [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) => 2
[[[],[]],[],[],[]] => [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) => 2
[[[],[],[],[],[]]] => [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) => 2
[[],[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[],[[]],[]] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[],[[],[]]] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[[]],[],[]] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[[]],[[]]] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[[],[]],[]] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[],[[],[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[[]],[],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[],[],[[]],[],[[]]] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
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Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
This is the minimal number of linear orderings whose intersection is the given poset.
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
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
This sends the maximal height of the Dyck path to the depth of the tree.
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