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Documents authored by Hwang, Hsien-Kuei

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Flajolet Lecture
DOI: 10.4230/LIPIcs.AofA.2026.1
Laplace, Cauchy and Early Analytic Combinatorics: Chance, Integrals, and Asymptotics (Flajolet Lecture)

Authors: Hsien-Kuei Hwang

Published in: LIPIcs, Volume 381, 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)

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.
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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)

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@InProceedings{hwang:LIPIcs.AofA.2026.1,
  author =	{Hwang, Hsien-Kuei},
  title =	{{Laplace, Cauchy and Early Analytic Combinatorics: Chance, Integrals, and Asymptotics}},
  booktitle =	{37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
  pages =	{1:1--1:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-435-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{381},
  editor =	{Panagiotou, Konstantinos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2026.1},
  URN =		{urn:nbn:de:0030-drops-262726},
  doi =		{10.4230/LIPIcs.AofA.2026.1},
  annote =	{Keywords: Laplace, Cauchy, analytic combinatorics, finite differences, saddle-point method, asymptotic analysis, integral transforms, occupancy problem}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.18
Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves

Authors: Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
The Colijn-Plazzotta ranking is a certain bijection between the unlabeled binary rooted trees and the positive integers, such that the integer associated with a tree is determined from the integers associated with the two immediate subtrees of its root. Letting a_n denote the minimal Colijn-Plazzotta rank among all trees with a specified number of leaves n, the sequence {a_n} begins 1, 2, 3, 4, 6, 7, 10, 11, 20, 22, 28, 29, 53, 56, 66, 67 (OEIS A354970). Here we show that a_n ∼ 2 [2^{P(log₂ n)}]ⁿ, where P varies as a periodic function dependent on {log₂ n} and satisfies 1.24602 < 2^{P(log₂ n)} < 1.33429.
Cite as

Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg. Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{doboli_et_al:LIPIcs.AofA.2024.18,
  author =	{Doboli, Michael R. and Hwang, Hsien-Kuei and Rosenberg, Noah A.},
  title =	{{Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.18},
  URN =		{urn:nbn:de:0030-drops-204530},
  doi =		{10.4230/LIPIcs.AofA.2024.18},
  annote =	{Keywords: Colijn-Plazzotta ranking, recurrences, unlabeled trees}
}
Document
DOI: 10.4230/LIPIcs.AofA.2018.13
Asymptotic Distribution of Parameters in Random Maps

Authors: Olivier Bodini, Julien Courtiel, Sergey Dovgal, and Hsien-Kuei Hwang

Published in: LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

Abstract
We consider random rooted maps without regard to their genus, with fixed large number of edges, and address the problem of limiting distributions for six different parameters: vertices, leaves, loops, root edges, root isthmus, and root vertex degree. Each of these leads to a different limiting distribution, varying from (discrete) geometric and Poisson distributions to different continuous ones: Beta, normal, uniform, and an unusual distribution whose moments are characterised by a recursive triangular array.
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Olivier Bodini, Julien Courtiel, Sergey Dovgal, and Hsien-Kuei Hwang. Asymptotic Distribution of Parameters in Random Maps. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bodini_et_al:LIPIcs.AofA.2018.13,
  author =	{Bodini, Olivier and Courtiel, Julien and Dovgal, Sergey and Hwang, Hsien-Kuei},
  title =	{{Asymptotic Distribution of Parameters in Random Maps}},
  booktitle =	{29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)},
  pages =	{13:1--13:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{Fill, James Allen and Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.13},
  URN =		{urn:nbn:de:0030-drops-89069},
  doi =		{10.4230/LIPIcs.AofA.2018.13},
  annote =	{Keywords: Random maps, Analytic combinatorics, Rooted Maps, Beta law, Limit laws, Patterns, Generating functions, Riccati equation}
}
Document
DOI: 10.4230/LIPIcs.AofA.2018.29
Asymptotic Expansions for Sub-Critical Lagrangean Forms

Authors: Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh

Published in: LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

Abstract
Asymptotic expansions for the Taylor coefficients of the Lagrangean form phi(z)=zf(phi(z)) are examined with a focus on the calculations of the asymptotic coefficients. The expansions are simple and useful, and we discuss their use in some enumerating sequences in trees, lattice paths and planar maps.
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Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh. Asymptotic Expansions for Sub-Critical Lagrangean Forms. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hwang_et_al:LIPIcs.AofA.2018.29,
  author =	{Hwang, Hsien-Kuei and Kang, Mihyun and Duh, Guan-Huei},
  title =	{{Asymptotic Expansions for Sub-Critical Lagrangean Forms}},
  booktitle =	{29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)},
  pages =	{29:1--29:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{Fill, James Allen and Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.29},
  URN =		{urn:nbn:de:0030-drops-89224},
  doi =		{10.4230/LIPIcs.AofA.2018.29},
  annote =	{Keywords: asymptotic expansions, Lagrangean forms, saddle-point method, singularity analysis, maps}
}
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