

























We study Dyck paths refined simultaneously by proper area and water capacity, where water capacity is measured above the path and below its lattice-path upper hull. The finite-height ingredients used in the enumeration are classical bounded-height area-polynomial and continued-fraction objects. The upper-hull decomposition produces a coupled area--capacity substitution, which gives an exact four-variable height expansion with denominator branches indexed by the height levels. The full generating function is asymmetric in the two weights, while the height summands admit a symmetric unreduced denominator representation under interchange of the area and capacity weights. In the open square $0<p,q<1$, we prove that the length radius of $G(x,1,p,q)$ is the minimum of the positive real denominator branches. The proof combines uniform normal convergence of the height expansion below this first branch with a Perron-root representation of the branch locations and an interval log-submodularity theorem for spectral radii of weighted paths. On the diagonal $p=q=s$, the classical Chebyshev specialisation gives explicit branch crossings and a $(1-s)^{2/3}$ branch-envelope accumulation law at the Dyck critical point.
此内容由惯性聚合(RSS阅读器)自动聚合整理,仅供阅读参考。 原文来自 — 版权归原作者所有。