Rigorous Asymptotic Analysis of 3-Noncrossing Skeleton Diagrams

Abstract

We give a complete rigorous asymptotic analysis of the generating functions of 3-noncrossing skeleton matchings and canonical 3-noncrossing skeleton diagrams. Let $F_3$ be the ordinary generating function of 3-noncrossing matchings, and let $S(y)=\sum_{n\geq 0}S(n)y^n$ be determined by $S(zF_3(z)^2)=F_3(z)$. The proof is deliberately ordered to avoid circularity. First, Lagrange inversion, a Stieltjes representation of $F_3$, exact cut-boundary estimates, and a moving horizontal Hankel contour give $S(n)\sim 24(\pi A^5)^{-1}\sigma^{-n}n^{-5}$ independently of any $\Delta$-analyticity of $S$. This estimate supplies boundary regularity of $S$ and $S'$. We then prove a global biholomorphic inversion theorem, continuation across every nonprincipal point of the convergence circle, and a logarithmically perturbed sectorial inverse theorem. A complete disk-chain and monodromy argument yields a single-valued continuation to a standard $\Delta$-domain. At the principal singularity, $S(y)=Q_4(u)-(\pi A^5)^{-1}u^4\log u+O(u^5(1+|\log u|))$, where $u=1-y/\sigma$. Finally, the canonical composition $S_3^{[4]}(z)=(1-z)(S(\vartheta(z))-1-\vartheta(z))$ is shown to be $\Delta$-analytic at its unique dominant singularity $\eta=0.49340718057613087519\ldots$, and $[z^n]S_3^{[4]}(z)\sim 7892.16205625817\ldots n^{-5}\eta^{-n}$. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.