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PreprintCC BY 4.0arXiv

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

  • Evan Conway
  • James Harbour
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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 √(πs(s+1)/8) · n3/2, while for fixed n it is asymptotic to sn · 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.

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Cite

BibTeXRIS
BibTeX
@misc{conway2026area,
  title = {{The Area Asymptotics of $(sn,n)$-Dyck Paths}},
  author = {Evan Conway and James Harbour},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07671},
  eprint = {2610.07671},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07671},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Area Asymptotics of (sn,n)-Dyck Paths
AU  - Evan Conway
AU  - James Harbour
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07671
UR  - https://arxiv.org/abs/2610.07671
AB  - 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 √(πs(s+1)/8) · n^3/2, while for fixed n it is asymptotic to sn · 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.
ER  -