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Laplace, Cauchy and Early Analytic Combinatorics: Chance, Integrals, and Asymptotics (Flajolet Lecture)

Author Hsien-Kuei Hwang

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Mathematical analysis
Keywords
  • Laplace
  • Cauchy
  • analytic combinatorics
  • finite differences
  • saddle-point method
  • asymptotic analysis
  • integral transforms
  • occupancy problem

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Abstract

This paper traces early historical developments of analytic combinatorics through a single object: the finite difference Δ^k 0ⁿ (the ordered Stirling numbers). We examine how Laplace transformed this discrete quantity into real integral representations to derive saddle-point approximations, establishing an early encoding-integration-approximation pipeline. Cauchy’s 1815 memoir then moved the same problem toward complex-analytic territory. Through this narrative we illustrate a pivotal transition: from an eighteenth-century algebra of formal identities to a nineteenth-century discipline of ε-δ inequalities.

Cite As Get BibTex

Hsien-Kuei Hwang. Laplace, Cauchy and Early Analytic Combinatorics: Chance, Integrals, and Asymptotics (Flajolet Lecture). In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.1

Author Details

Hsien-Kuei Hwang
  • Institute of Statistical Science, Academia Sinica, Taipei, Taiwan

Funding

The research was partially supported by the NSTC, Grant NSTC-114-2118-M-001-005-MY3.

Acknowledgements

The author acknowledges the use of AI-assisted tools (ChatGPT, Gemini, and Claude) for exploratory discussion, language polishing, LaTeX editing, and consistency checks. These tools were not used as authorities for historical or mathematical claims; all such claims and references were independently verified by the author, who bears sole responsibility for the content of the paper.

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