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DOI: 10.4230/LIPIcs.AofA.2026.13
A Combinatorial Framework for the Pons-Batle Identity: Young Tableaux, Lattice Paths, and Limit Laws

Authors: Hexuan Liu, Michael Wallner, and Guan-Ru Yu

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

Abstract
Tree-child networks are an important class of phylogenetic network used to model reticulate evolutionary processes. These networks have attracted increasing attention from researchers with interests in both combinatorics and algorithms. A fundamental open problem posed by Pons and Batle asks whether the number TC_{n,k} of bicombining tree-child networks with n leaves and k reticulation nodes equals the number of certain constrained words, now called Pons-Batle words. In this paper, we confirm the conjecture for tree-child networks with a bounded number of reticulation nodes. Our approach is combinatorial and analytic. We introduce families of Young tableaux with walls and holes and construct explicit bijections with Pons-Batle words, yielding a direct combinatorial explanation of the identities. These tableaux encode structural features of the underlying networks, including the placement of reticulation nodes. By projecting them to decorated Dyck paths, we obtain algebraic generating functions with differential operators encoding step weights, leading to explicit recurrence relations and closed-form formulas for TC_{n,k}. Beyond finite verification for moderate k, the framework reveals an underlying probabilistic structure. For k = 1, natural structural parameters, such as the position and value of distinguished cells, converge, after rescaling, to Beta(2,1), Beta(1,2), and Uniform (i.e., Beta(1,1)) distributions. These limit laws arise from a coalescence of singularities at the dominant square-root singularity, producing a non-analytic transition in the local expansion. Overall, our results provide both combinatorial insight and a unified analytic perspective on the asymptotic behavior of tree-child networks, showing how algebraic generating functions with interacting singularities systematically produce Beta limit laws.
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Hexuan Liu, Michael Wallner, and Guan-Ru Yu. A Combinatorial Framework for the Pons-Batle Identity: Young Tableaux, Lattice Paths, and Limit Laws. 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. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{liu_et_al:LIPIcs.AofA.2026.13,
  author =	{Liu, Hexuan and Wallner, Michael and Yu, Guan-Ru},
  title =	{{A Combinatorial Framework for the Pons-Batle Identity: Young Tableaux, Lattice Paths, and Limit Laws}},
  booktitle =	{37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
  pages =	{13:1--13:20},
  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.2026.13},
  URN =		{urn:nbn:de:0030-drops-262848},
  doi =		{10.4230/LIPIcs.AofA.2026.13},
  annote =	{Keywords: Recurrence relations, generating functions, analytic combinatorics, Young tableaux with walls, constrained words, bijections, exact enumeration}
}
Document
DOI: 10.4230/LIPIcs.AofA.2022.5
Enumeration of d-Combining Tree-Child Networks

Authors: Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu

Published in: LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

Abstract
Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to d-combining tree-child networks where every reticulation node has now d ≥ 2 parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves.
Cite as

Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumeration of d-Combining Tree-Child Networks. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chang_et_al:LIPIcs.AofA.2022.5,
  author =	{Chang, Yu-Sheng and Fuchs, Michael and Liu, Hexuan and Wallner, Michael and Yu, Guan-Ru},
  title =	{{Enumeration of d-Combining Tree-Child Networks}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-230-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{225},
  editor =	{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.2022.5},
  URN =		{urn:nbn:de:0030-drops-160914},
  doi =		{10.4230/LIPIcs.AofA.2022.5},
  annote =	{Keywords: Phylogenetic network, tree-child network, d-combining tree-child network, exact enumeration, asymptotic enumeration, reticulation node, limit law, stretched exponential}
}
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