ThreeCatalanPathCount on DyckPath

The paths from the origin to (n, n, n) with x >= y >= z whose first two coordinates trace this Dyck path and whose last two coordinates trace it as well.

On DyckPath

With r the downs between consecutive up steps and s the ups between consecutive down steps, the product over consecutive pairs of binomial(r + s, r).

Called as CombinatorialStat(x, "ThreeCatalanPathCount"). The head's page has its examples.

Live

Distribution

Over DyckPaths at each size, tallied here in your browser; sizes past 10 are not offered.

Sizes 1 to

FindStat St001800

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections.

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$. Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by proj…

Object	Value
[]	1
[1,0]	1
[1,0,1,0]	2
[1,1,0,0]	1
[1,0,1,0,1,0]	4
[1,0,1,1,0,0]	3
[1,1,0,0,1,0]	3
[1,1,0,1,0,0]	1

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001800 has the rest.