- FindStat

Identifier

Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000069: 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) => 1

[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) => 1

[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

[1,1,0,1,0,1,0,0] => ([(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) => 1

[1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1

[1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1

[1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1

[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) => 1

[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

[1,0,1,1,0,1,0,1,0,0] => ([(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) => 1

[1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[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) => 1

[1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[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) => 1

[1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,1,0,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1

[1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1

[1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1

[1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1

[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) => 1

[1,1,1,0,1,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 1

[1,1,1,0,1,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 1

[1,1,1,0,1,0,1,0,0,0] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1

[1,1,1,0,1,1,0,0,0,0] => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8) => 1

[1,1,1,1,0,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 1

[1,1,1,1,0,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 1

[1,1,1,1,0,0,1,0,0,0] => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8) => 1

[1,1,1,1,0,1,0,0,0,0] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1

[1,1,1,1,1,0,0,0,0,0] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1

[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

[1,0,1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 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

[1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 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) => 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) => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 1

[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

[1,0,1,1,0,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 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

[1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 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) => 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) => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 1

[1,0,1,1,1,0,0,1,0,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8) => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9) => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9) => 1

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 5\ q$

$F_{4} = 14\ q$

$F_{5} = 42\ q$

$F_{6} = 132\ q$

$F_{7} = 429\ q$

Description

The number of maximal elements of a 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.