- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001880: Posets ⟶ ℤ

Values

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

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

click to show known generating functions

$F_{2} = q^{3} + q^{4}$

$F_{3} = q^{4} + 2\ q^{5} + 2\ q^{6}$

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

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.