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 F3 be the ordinary generating function of 3-noncrossing matchings, and let S(y)=∑n≥ 0S(n)yn be determined by S(zF3(z)2)=F3(z). The proof is deliberately ordered to avoid circularity. First, Lagrange inversion, a Stieltjes representation of F3, exact cut-boundary estimates, and a moving horizontal Hankel contour give S(n)∼ 24(πA5)−1σ−nn−5 independently of any Δ-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 Δ-domain. At the principal singularity, S(y)=Q4(u)−(πA5)−1u4log u+O(u5(1+|log u|)), where u=1−y/σ. Finally, the canonical composition S3[4](z)=(1−z)(S(ϑ(z))−1−ϑ(z)) is shown to be Δ-analytic at its unique dominant singularity η=0.49340718057613087519…, and [zn]S3[4](z)∼ 7892.16205625817… n−5η−n. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.
Related
- The quasisymmetric flag variety: a toric complex on noncrossing partitions
- Categorifications of noncrossing partitions of type B and affine type A
- Noncrossing combinatorics of quasi-Coxeter elements
- Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms
- Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups
BibTeX
@misc{zhao2026rigorous,
title = {{Rigorous Asymptotic Analysis of 3-Noncrossing Skeleton Diagrams}},
author = {Yangyang Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.24425},
eprint = {2608.24425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24425},
}
RIS
TY - UNPB
TI - Rigorous Asymptotic Analysis of 3-Noncrossing Skeleton Diagrams
AU - Yangyang Zhao
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.24425
UR - https://arxiv.org/abs/2608.24425
AB - 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)=∑_n≥ 0S(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)∼ 24(πA^5)^−1σ^−nn^−5 independently of any Δ-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 Δ-domain. At the principal singularity, S(y)=Q_4(u)−(πA^5)^−1u^4log u+O(u^5(1+|log u|)), where u=1−y/σ. Finally, the canonical composition S_3^[4](z)=(1−z)(S(ϑ(z))−1−ϑ(z)) is shown to be Δ-analytic at its unique dominant singularity η=0.49340718057613087519…, and [z^n]S_3^[4](z)∼ 7892.16205625817… n^−5η^−n. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.
ER -