The Area Asymptotics of $(sn,n)$-Dyck Paths

Abstract

We study the total and average area of $(sn,n)$-Dyck paths: lattice paths from $(0,0)$ to $(sn,n)$ that stay weakly below the line $x=sy$, counted by the Fuss-Catalan numbers. Generalizing a result of Merlini, Sprugnoli, and Verri for the case $s=1$, we derive an exact formula for the total area over all such paths. From this we obtain explicit upper and lower bounds for both the total and the average area, together with the corresponding asymptotics: for fixed $s$, the average area is asymptotic to $\sqrt{\pi s(s+1)/8}\cdot n^{3/2}$, while for fixed $n$ it is asymptotic to $sn\cdot Q(n)/2$ as $s$ grows large, where $Q(n)$ denotes Ramanujan's $Q$-function. Along the way, we confirm a conjecture of Kotesovec on the asymptotics of a binomial sum that also arises in several other enumeration problems.

arXiv comment21 pages, 4 figures