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

Galled trees appear in problems concerning admixture, horizontal gene transfer, hybridization, and recombination. Building on a recursive enumerative construction, we study the asymptotic behavior of the number of rooted binary unlabeled (normal) galled trees as the number of leaves n increases, maintaining a fixed number of galls g. We find that the exponential growth with n of the number of rooted binary unlabeled normal galled trees with g galls has the same value irrespective of the value of g ≥ 0. The subexponential growth, however, depends on g; it follows c_g n^{2g-3/2}, where c_g is a constant dependent on g. Although for each g, the exponential growth is approximately 2.4833ⁿ, summing across all g, the exponential growth is instead approximated by the much larger 4.8230ⁿ.

Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls. 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. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agranattamir_et_al:LIPIcs.AofA.2024.27, author = {Agranat-Tamir, Lily and Fuchs, Michael and Gittenberger, Bernhard and Rosenberg, Noah A.}, title = {{Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {27:1--27:14}, 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.2024.27}, URN = {urn:nbn:de:0030-drops-204626}, doi = {10.4230/LIPIcs.AofA.2024.27}, annote = {Keywords: galled trees, generating functions, phylogenetics, unlabeled trees} }

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

In a companion article dedicated to the enumeration aspects, we showed how to obtain closed form formulas for the generating functions of walks, bridges, meanders, and excursions avoiding any fixed word (a pattern p). The autocorrelation polynomial of this forbidden pattern p (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) plays a crucial role. In this article, we get the asymptotics of these walks. We also introduce a trivariate generating function (length, final altitude, number of occurrences of p), for which we derive a closed form. We prove that the number of occurrences of p is normally distributed: This is what Flajolet and Sedgewick call an instance of Borges's theorem.
We thus extend and refine the study by Banderier and Flajolet in 2002 on lattice paths, and we unify several dozens of articles which investigated patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. Our approach relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. The situation is much more involved than in the Banderier-Flajolet work: forbidden patterns lead to a wider zoology of asymptotic behaviours, and we classify them according to the geometry of a Newton polygon associated with these constrained walks, and we analyse what are the universal phenomena common to all these models of lattice paths avoiding a pattern.

Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger. Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem. 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. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{asinowski_et_al:LIPIcs.AofA.2018.10, author = {Asinowski, Andrei and Bacher, Axel and Banderier, Cyril and Gittenberger, Bernhard}, title = {{Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {10:1--10:14}, 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.2018.10}, URN = {urn:nbn:de:0030-drops-89035}, doi = {10.4230/LIPIcs.AofA.2018.10}, annote = {Keywords: Lattice paths, pattern avoidance, finite automata, context-free languages, autocorrelation, generating function, kernel method, asymptotic analysis, Gaussian limit law} }

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

We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, and by a bound of the nesting levels of abstractions, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable.
We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to C_1 n and C_2 n, respectively, where the constants C_1 and C_2 depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with a bounded number of abstractions between each variable and its binding lambda. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of nesting levels of abstractions, we investigate the number of variables in the different abstraction levels and thereby exhibit very interesting results concerning the distribution of the variables within those lambda-terms.

Bernhard Gittenberger and Isabella Larcher. On the Number of Variables in Special Classes of Random Lambda-Terms. 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. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gittenberger_et_al:LIPIcs.AofA.2018.25, author = {Gittenberger, Bernhard and Larcher, Isabella}, title = {{On the Number of Variables in Special Classes of Random Lambda-Terms}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {25:1--25:14}, 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.2018.25}, URN = {urn:nbn:de:0030-drops-89180}, doi = {10.4230/LIPIcs.AofA.2018.25}, annote = {Keywords: lambda-terms, directed acyclic graphs, singularity analysis, limiting distributions} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of 0-1-strings. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n) is o(n^{-3/2} tau^{-n}) for tau ~ 1.963448... . We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Theta(n^{-3/2} * rho^{-n}) with some class dependent constant rho, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.

Bernhard Gittenberger and Zbigniew Golebiewski. On the Number of Lambda Terms With Prescribed Size of Their De Bruijn Representation. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gittenberger_et_al:LIPIcs.STACS.2016.40, author = {Gittenberger, Bernhard and Golebiewski, Zbigniew}, title = {{On the Number of Lambda Terms With Prescribed Size of Their De Bruijn Representation}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {40:1--40:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.40}, URN = {urn:nbn:de:0030-drops-57411}, doi = {10.4230/LIPIcs.STACS.2016.40}, annote = {Keywords: lambda calculus, terms enumeration, analytic combinatorics} }

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