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LIPIcs, Volume 381
37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)
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- published at: 2026-07-13
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-435-2
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Complete Volume
LIPIcs, Volume 381, AofA 2026, Complete Volume
Abstract
LIPIcs, Volume 381, AofA 2026, Complete Volume
Cite as
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. 1-520, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{panagiotou:LIPIcs.AofA.2026,
title = {{LIPIcs, Volume 381, AofA 2026, Complete Volume}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {1--520},
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},
URN = {urn:nbn:de:0030-drops-270170},
doi = {10.4230/LIPIcs.AofA.2026},
annote = {Keywords: LIPIcs, Volume 381, AofA 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
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. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{panagiotou:LIPIcs.AofA.2026.0,
author = {Panagiotou, Konstantinos},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {0:i--0:xiv},
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.2026.0},
URN = {urn:nbn:de:0030-drops-270164},
doi = {10.4230/LIPIcs.AofA.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Flajolet Lecture
Laplace, Cauchy and Early Analytic Combinatorics: Chance, Integrals, and Asymptotics (Flajolet Lecture)
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
Path Length and External Path Length in Random Trees
Abstract
We consider two closely related concepts in rooted trees: the path length is the sum of all distances from the root to the vertices of the tree, while the external path length is the sum of all distances from the root to the leaves of the tree. Upon dividing by the number of vertices and leaves respectively, we obtain the average distance to the root. For two important classes of random trees, we show that the average distance of the root to a random leaf is almost the same as the average distance to a random vertex. For Bienaymé-Galton-Watson trees, the difference is bounded in probability. For three varieties of increasing trees (recursive trees, d-ary increasing trees, generalised plane-oriented recursive trees), on the other hand, the difference converges to a constant in probability.
Cite as
Jacob Lundblad and Stephan Wagner. Path Length and External Path Length in Random Trees. 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. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lundblad_et_al:LIPIcs.AofA.2026.2,
author = {Lundblad, Jacob and Wagner, Stephan},
title = {{Path Length and External Path Length in Random Trees}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {2:1--2:20},
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.2026.2},
URN = {urn:nbn:de:0030-drops-262733},
doi = {10.4230/LIPIcs.AofA.2026.2},
annote = {Keywords: Path length, external path length, Bienaym\'{e}-Galton-Watson trees, increasing trees}
}
Document
Efficient Sampling of Increasing Trees
Abstract
We present a new exact-size sampler for increasing trees that outputs a tree of size n uniformly at random while avoiding the global coefficient pre-computation required by the classical recursive method of Flajolet et al. [Philippe Flajolet et al., 1994]. The key idea is a hybrid oracle-driven rejection scheme in which local sampling decisions are made using interval bounds on the coefficients, with a fallback to exact recurrence computation only on rare ambiguous events. In the bit-complexity model this yields an expected running time of O(nlog n) and it consumes a number of random bits within O(n) of the Shannon entropy, which is information-theoretically optimal up to lower-order terms. The sampler proceeds in two phases. We first generate the unlabeled rooted ordered shape by recursively sampling node arities and subtree sizes and then draw a uniform permutation of {1,…,n} and apply a deterministic increasing-labeling procedure.
Cite as
Nadja Azzouz, Olivier Bodini, Francis Durand, and Bernhard Gittenberger. Efficient Sampling of Increasing Trees. 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. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azzouz_et_al:LIPIcs.AofA.2026.3,
author = {Azzouz, Nadja and Bodini, Olivier and Durand, Francis and Gittenberger, Bernhard},
title = {{Efficient Sampling of Increasing Trees}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {3:1--3:16},
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.2026.3},
URN = {urn:nbn:de:0030-drops-262740},
doi = {10.4230/LIPIcs.AofA.2026.3},
annote = {Keywords: sampling algorithms, bit-complexity, increasing trees, generating functions}
}
Document
Scaling Limits of Multitype Bienaymé Trees
Abstract
We first consider irreducible critical multitype Bienaymé trees and extend the results to the case, when they possess a critical irreducible component with attached subcritical components. We study these trees under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong non-asymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.
Cite as
Louigi Addario-Berry, Philipp Beltran, Benedikt Stufler, and Paul Thévenin. Scaling Limits of Multitype Bienaymé Trees. 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. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{addarioberry_et_al:LIPIcs.AofA.2026.4,
author = {Addario-Berry, Louigi and Beltran, Philipp and Stufler, Benedikt and Th\'{e}venin, Paul},
title = {{Scaling Limits of Multitype Bienaym\'{e} Trees}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {4:1--4:16},
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.2026.4},
URN = {urn:nbn:de:0030-drops-262750},
doi = {10.4230/LIPIcs.AofA.2026.4},
annote = {Keywords: branching processes, multitype trees, scaling limit}
}
Document
Semi-Simplex Phylogenetic Networks: Tree-Child Networks and Galled Trees
Abstract
Understanding the size of phylogenetic network classes and the typical shape of a random network from a fixed class has been one of the major research focuses in phylogenetics over the last couple of years. In this extended abstract, we consider two subclasses of the (recently introduced) class of semi-simplex phylogenetic networks, namely, semi-simplex tree-child networks and semi-simplex galled trees. We clarify their sizes relative to the (known) sizes of general tree-child networks and galled trees, respectively, and prove limit laws for parameters of random networks from these classes. Additional classes of semi-simplex networks will be considered in the journal version of this paper.
Cite as
Michael Fuchs and Tsan-Cheng Yu. Semi-Simplex Phylogenetic Networks: Tree-Child Networks and Galled Trees. 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. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fuchs_et_al:LIPIcs.AofA.2026.5,
author = {Fuchs, Michael and Yu, Tsan-Cheng},
title = {{Semi-Simplex Phylogenetic Networks: Tree-Child Networks and Galled Trees}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {5:1--5:17},
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.2026.5},
URN = {urn:nbn:de:0030-drops-262768},
doi = {10.4230/LIPIcs.AofA.2026.5},
annote = {Keywords: Semi-simplex networks, tree-child networks, galled trees, enumeration, limit laws, Sackin index}
}
Document
Local Central Limit Theorems for Subgraph Counts in Subcritical Graph Families
Abstract
It was already established in [Drmota et al., 2017] that subgraph counts in vertex labelled subcritial graph families satisfy a central limit theorem. This result is now sharpened to local central limit theorems. Furthermore this result is generalized to unlabelled subcritical graph families and to multivariate central limit theorems for the joint distribution of finitely many subgraph counts.
Cite as
Michael Drmota and Yitian Wang. Local Central Limit Theorems for Subgraph Counts in Subcritical Graph Families. 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. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{drmota_et_al:LIPIcs.AofA.2026.6,
author = {Drmota, Michael and Wang, Yitian},
title = {{Local Central Limit Theorems for Subgraph Counts in Subcritical Graph Families}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {6:1--6:13},
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.2026.6},
URN = {urn:nbn:de:0030-drops-262779},
doi = {10.4230/LIPIcs.AofA.2026.6},
annote = {Keywords: Subcritical graph classes, subgraph counts, generating functions, functional equations}
}
Document
Ancestries and Descendants in a Random DAG
Abstract
We consider a random recursive DAG G_n on the vertex set [n] where every vertex i ≥ 2 has out-degree d, with the targets chosen uniformly at random among the earlier i-1 vertices. For this model, we propose a novel way to investigate the descendants of n (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process a_i(n) of a vertex i is defined as the number of ancestors of i in G_n, and is closely related to the evolutions of multi-draw Pólya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. We employ this method to make progress on two open problems posed by Janson, as well as to provide an alternative proof of a first-moment result contained in his work. We further prove limit theorems for the ancestor processes a_i(n) depending on i.
Cite as
Fabian Burghart. Ancestries and Descendants in a Random DAG. 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. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{burghart:LIPIcs.AofA.2026.7,
author = {Burghart, Fabian},
title = {{Ancestries and Descendants in a Random DAG}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {7:1--7:14},
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.2026.7},
URN = {urn:nbn:de:0030-drops-262785},
doi = {10.4230/LIPIcs.AofA.2026.7},
annote = {Keywords: Random DAG, descendants, Markov process, Urn model, Limit theorems}
}
Document
Local Limit of Random Regular Bipartite Planar Maps
Abstract
We prove the existence of the local limit of uniform random d-regular bipartite planar maps, for every d ≥ 3, as the number of vertices tends to infinity. The proof relies on a bijection between maps and so-called blossoming trees established in a previous work. After proving local convergence of the associated decorated trees, we extend the bijection to infinite trees and transfer the convergence to planar maps. The limiting object is almost surely one-ended and recurrent for the simple random walk.
Cite as
Nicolas Tokka. Local Limit of Random Regular Bipartite Planar Maps. 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. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tokka:LIPIcs.AofA.2026.8,
author = {Tokka, Nicolas},
title = {{Local Limit of Random Regular Bipartite Planar Maps}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {8:1--8:19},
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.2026.8},
URN = {urn:nbn:de:0030-drops-262791},
doi = {10.4230/LIPIcs.AofA.2026.8},
annote = {Keywords: Planar maps, random maps and trees, local convergence}
}
Document
Link Between Bipartite and General Unicellular Toroidal Maps via Slit-Slide-Sew Bijections
Abstract
We relate general maps to bipartite maps through a bijection of type slit-slide-sew. We provide an involution on arbitrary genus maps with even degree faces. This enables a full interpretation of the relation between general and bipartite maps, in the case of genus 1 maps with a unique face.
The main tool is the use of rotations along well-chosen specific loops. Once a noncontractible simple loop is given, one slits along it, slides one notch, and sews back. This mildly modifies the structure of the map along the loop, changing the parity of the length of other loops crossing it. In the unicellular toroidal setting, the structure of noncontractible loops is simple enough to enable a full correspondence between general and bipartite maps.
Cite as
Jérémie Bettinelli and Dimitri Korkotashvili. Link Between Bipartite and General Unicellular Toroidal Maps via Slit-Slide-Sew Bijections. 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. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bettinelli_et_al:LIPIcs.AofA.2026.9,
author = {Bettinelli, J\'{e}r\'{e}mie and Korkotashvili, Dimitri},
title = {{Link Between Bipartite and General Unicellular Toroidal Maps via Slit-Slide-Sew Bijections}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {9:1--9:17},
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.2026.9},
URN = {urn:nbn:de:0030-drops-262804},
doi = {10.4230/LIPIcs.AofA.2026.9},
annote = {Keywords: combinatorial map, enumeration, bijection, unicellular map, slit-slide-sew}
}
Document
Gibbs Partitions and Lattice Paths
Abstract
We study a Gibbs partition model (composition scheme) under a new condition on the component weights, leading to a previously unobserved regime for the number of components. We establish a condensation phenomenon producing a unique giant component, and prove a Cox process limit describing a sublinear power-law growth of sizes of non-maximal components. Our results are motivated by applications to lattice paths and random walks, including simple random walks in the cube, Delannoy paths, pairs of Dyck bridges, urn models and card guessing games.
Cite as
Niccolò Bosio, Markus Kuba, and Benedikt Stufler. Gibbs Partitions and Lattice Paths. 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. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bosio_et_al:LIPIcs.AofA.2026.10,
author = {Bosio, Niccol\`{o} and Kuba, Markus and Stufler, Benedikt},
title = {{Gibbs Partitions and Lattice Paths}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {10:1--10:12},
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.2026.10},
URN = {urn:nbn:de:0030-drops-262814},
doi = {10.4230/LIPIcs.AofA.2026.10},
annote = {Keywords: Gibbs partitions, composition schemes, lattice paths, random walks, condensation}
}
Document
Poisson-Dirichlet Graphons and Permutons
Abstract
We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a heavy-tailed regime and establish a comprehensive phase diagram for the asymptotic shape of superstructures.
Cite as
Benedikt Stufler. Poisson-Dirichlet Graphons and Permutons. 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. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{stufler:LIPIcs.AofA.2026.11,
author = {Stufler, Benedikt},
title = {{Poisson-Dirichlet Graphons and Permutons}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {11:1--11:16},
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.2026.11},
URN = {urn:nbn:de:0030-drops-262821},
doi = {10.4230/LIPIcs.AofA.2026.11},
annote = {Keywords: Graphons, Permutons, Poisson-Dirichlet point processes}
}
Document
Asymptotic Transfer in Critical Recursive Composition Schemes
Abstract
The composition ℱ∘𝒢 of two combinatorial classes ℱ and 𝒢 is a standard combinatorial construction and translates into the composition F(G(z)) of their corresponding counting generating functions. Such a composition is called critical if G(ρ_G) = ρ_F, where ρ_F and ρ_G denote the corresponding radii of convergence of F and G, respectively. In this case, both the singular behaviours of F and G influence that of F∘G. Such critical composition schemes arise frequently in map enumeration. For example, by using the block-decomposition, one has M(z) = B (z(1+M(z))²) and ρ_B = ρ_M (1+M(ρ_M))², where M(z) denotes the generating function of all rooted planar maps and B(y) the generating functions of 2-connected rooted planar maps. This can be extended to multivariate generating functions by taking several statistics into account, for example face counts. Since critical composition schemes exhibit (usually) a condensation phenomenon - in the above situation this means that there is a giant 2-connected block of linear size and linearly many small blocks - it is very plausible that statistical properties on 2-connected maps transfer to corresponding properties of all maps and back. The purpose of the present paper is to make this precise at the level of the singular structure of the corresponding multivariate generating functions. In particular, we show that moving 3/2-singularities transfer. Since such singularities are closely related to central limit theorems of the corresponding statistics, this method also provides a kind of transfer of central limit theorems. Actually, this method is quite flexible and is applied to a variety of face and pattern counting statistics in map enumeration.
Cite as
Michael Drmota and Zéphyr Salvy. Asymptotic Transfer in Critical Recursive Composition Schemes. 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. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{drmota_et_al:LIPIcs.AofA.2026.12,
author = {Drmota, Michael and Salvy, Z\'{e}phyr},
title = {{Asymptotic Transfer in Critical Recursive Composition Schemes}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {12:1--12:17},
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.2026.12},
URN = {urn:nbn:de:0030-drops-262834},
doi = {10.4230/LIPIcs.AofA.2026.12},
annote = {Keywords: Analytic Combinatorics, Central Limit Theorem, Pattern Counts, Random Planar Maps, Singularity Analysis}
}
Document
A Combinatorial Framework for the Pons-Batle Identity: Young Tableaux, Lattice Paths, and Limit Laws
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.
Cite as
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
The Cost of Cyclic Permutations and Remainder Sums in the Euclidean Algorithm
Abstract
We discuss a modification to the Gries-Mills block swapping scheme for in-place rotation with average costs of 1.85 moves per element and worst case performance still at 3 moves per element. The analysis of the average case relies on the asymptotic behavior of the sum of remainders in the Euclidean algorithm.
Cite as
Valentin Blomer and Kai-Uwe Bux. The Cost of Cyclic Permutations and Remainder Sums in the Euclidean Algorithm. 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. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blomer_et_al:LIPIcs.AofA.2026.14,
author = {Blomer, Valentin and Bux, Kai-Uwe},
title = {{The Cost of Cyclic Permutations and Remainder Sums in the Euclidean Algorithm}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {14:1--14:17},
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.2026.14},
URN = {urn:nbn:de:0030-drops-262858},
doi = {10.4230/LIPIcs.AofA.2026.14},
annote = {Keywords: generic algorithms, average case analysis, Euclidean algorithm, exponential sums, cyclic permutations}
}
Document
Cycle Structure of Random Standardized Permutations
Abstract
In this article, we study a model of random permutations, which we call random standardized permutations, and which is based on a sequence of i.i.d. random variables. This model generalizes other ones, such as the riffle shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. We prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then, we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally, we prove a central limit theorem for the total number of cycles.
Cite as
Aurélien Guerder. Cycle Structure of Random Standardized Permutations. 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. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guerder:LIPIcs.AofA.2026.15,
author = {Guerder, Aur\'{e}lien},
title = {{Cycle Structure of Random Standardized Permutations}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {15:1--15:16},
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.2026.15},
URN = {urn:nbn:de:0030-drops-262864},
doi = {10.4230/LIPIcs.AofA.2026.15},
annote = {Keywords: Random permutations, primitive words, combinatorics, asymptotic distribution, method of moments}
}
Document
On Cycles in Multiset Permutations, Parking Functions, and Related Structures
Abstract
In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.
Cite as
Calum Buchanan, Fabian Burghart, Stephan Wagner, and Mei Yin. On Cycles in Multiset Permutations, Parking Functions, and Related Structures. 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. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{buchanan_et_al:LIPIcs.AofA.2026.16,
author = {Buchanan, Calum and Burghart, Fabian and Wagner, Stephan and Yin, Mei},
title = {{On Cycles in Multiset Permutations, Parking Functions, and Related Structures}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {16:1--16:17},
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.2026.16},
URN = {urn:nbn:de:0030-drops-262874},
doi = {10.4230/LIPIcs.AofA.2026.16},
annote = {Keywords: parking function, multiset permutation, cycle type, cyclic point, terminal closer, equivalence of ensembles}
}
Document
Singularly Perturbed Discrete Differential Equations and Pattern Counts in Simple Triangulations
Abstract
Discrete differential equations of order k are of the form R(z,u,F(z,u),Δ F(z,u),…,Δ^kF(z,u)) = 0, where Δ F(z,u) = (F(z,u)-F(z,0))/u and Δ^k F(z,u) = Δ(Δ^{k-1} F(z,u)) for k ≥ 2. Such equations appear most prominently in planar map enumeration but also in several other contexts such as statistical mechanics, lattice path enumeration, pattern avoiding permutations or stack-sortable permutations. Mostly, one is interested in the function F(z,0) that is usually the corresponding counting generating function. In this work, we consider discrete differential equations with an additional parameter x, where the order of the equation is 1 for x = 1 but k > 1 for x ≠ 1. We call such equations singularly perturbed. The solution theory of higher order discrete differential equations is much more involved than for degree 1 and it is a priori not clear that there is a smooth transition from x = 1 to x ≠ 1. The main contribution of this work is to show that there is actually a smooth transition under certain natural assumptions. As an application of this result we consider pattern counts in triangular planar maps and derive a central limit theorem for these counts.
Cite as
Michael Drmota and Eva-Maria Hainzl. Singularly Perturbed Discrete Differential Equations and Pattern Counts in Simple Triangulations. 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. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{drmota_et_al:LIPIcs.AofA.2026.17,
author = {Drmota, Michael and Hainzl, Eva-Maria},
title = {{Singularly Perturbed Discrete Differential Equations and Pattern Counts in Simple Triangulations}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {17:1--17:14},
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.2026.17},
URN = {urn:nbn:de:0030-drops-262880},
doi = {10.4230/LIPIcs.AofA.2026.17},
annote = {Keywords: Discrete differential equations, catalytic equations, generating functions}
}
Document
Large Deviation Principles for Pattern-Avoiding Permutations, and Limit Shapes for Constrained Mallows Permutations
Abstract
We study Mallows random permutations conditioned to avoid a given pattern α of length 3, for which we find limit shapes in the space of permutons when the bias parameter is of the form e^(β/n). Along the way, we provide parametrizations for α-avoiding permutons, and establish large deviation principles for uniform α-avoiding permutations.
Cite as
Thomas Budzinski, Victor Dubach, Valentin Féray, Mohamed Slim Kammoun, and Mylene Maïda. Large Deviation Principles for Pattern-Avoiding Permutations, and Limit Shapes for Constrained Mallows Permutations. 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. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{budzinski_et_al:LIPIcs.AofA.2026.18,
author = {Budzinski, Thomas and Dubach, Victor and F\'{e}ray, Valentin and Kammoun, Mohamed Slim and Ma\"{i}da, Mylene},
title = {{Large Deviation Principles for Pattern-Avoiding Permutations, and Limit Shapes for Constrained Mallows Permutations}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {18:1--18:16},
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.2026.18},
URN = {urn:nbn:de:0030-drops-262895},
doi = {10.4230/LIPIcs.AofA.2026.18},
annote = {Keywords: Permutations, Pattern avoidance, Large deviations, Mallows distribution}
}
Document
Fringe Subtrees of Split Trees
Abstract
We consider additive functionals X_n(ϕ) with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if all balls are distributed multinomially and for some models with Beta-distributed splitter. Generally, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.
Cite as
Cecilia Holmgren, Jasper Ischebeck, and Svante Janson. Fringe Subtrees of Split Trees. 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. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{holmgren_et_al:LIPIcs.AofA.2026.19,
author = {Holmgren, Cecilia and Ischebeck, Jasper and Janson, Svante},
title = {{Fringe Subtrees of Split Trees}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {19:1--19:17},
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.2026.19},
URN = {urn:nbn:de:0030-drops-262900},
doi = {10.4230/LIPIcs.AofA.2026.19},
annote = {Keywords: fringe tree, split tree, fractional split tree, limit theorem, additive functional, renewal theory, Fourier series}
}
Document
Formulas and Asymptotics of Hypergraph Catalan Numbers
Abstract
Tree walks are a class of closed walks on a complete graph constrained to span trees. They appear in the computation of moments of the spectral measure of various random matrix models, most prominently in the spectral distribution of random graphs. In this work, we focus on a special subclass called k-tours, which were introduced by Gunnells [Gunnels, 2021] after studying another random matrix model. They are enumerated by the so-called hypergraph Catalan numbers c_n^(k). Gunnells conjectured an asymptotic formula for c_n^(k), which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of k-tours on star-like trees.
Cite as
Eva-Maria Hainzl. Formulas and Asymptotics of Hypergraph Catalan Numbers. 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. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hainzl:LIPIcs.AofA.2026.20,
author = {Hainzl, Eva-Maria},
title = {{Formulas and Asymptotics of Hypergraph Catalan Numbers}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {20:1--20:15},
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.2026.20},
URN = {urn:nbn:de:0030-drops-262918},
doi = {10.4230/LIPIcs.AofA.2026.20},
annote = {Keywords: generating functions, trees, tree walks, catalan numbers}
}
Document
A New Fine-Scale Berry-Esseen-Type Gumbel-Limit Theorem for Multivariate Maxima
Abstract
For d ≥ 2 and i.i.d. d-dimensional observations 𝐗^(1), 𝐗^(2), … with independent Exponential(1) coordinates, let φ_n denote the minimum 𝓁¹-norm among the maxima of {𝐗^(1), …, 𝐗^(n)}. (A maximum from this set is an observation 𝐗^(k) with 1 ≤ k ≤ n such that 𝐗^(k) ⊀ 𝐗^(i) for all 1 ≤ i ≤ n, where 𝐱 ≺ 𝐲 means that x_j < y_j for 1 ≤ j ≤ d.) Key roles in the study of multivariate Pareto records are played by φ_n and by the more easily handled maximum with the maximum 𝓁¹-norm. Fill et al. proved [Fill et al., 2026, Theorem 1.11(a)] that φ_n = ln n - ln ln ln n - ln(d - 1) + O_p(1/ln ln n), where Z_n = O_p(a_n) means that Z_n / a_n is bounded in probability, and conjectured [Fill et al., 2026, Remarks 1.13 and 3.3] that (ln ln n) (φ_n - [ln n - ln ln ln n - ln(d - 1)]) has a nondegenerate limiting distribution, suggesting that the limiting distribution might be that of - G, where G has a Gumbel distribution with location - ln[(d - 1)!]/(d - 1) and scale 1/(d - 1). In the present extended abstract we outline a proof of a Berry-Esseen-type theorem for this convergence in distribution, thereby establishing a very sharp result for φ_n.
Cite as
James Allen Fill. A New Fine-Scale Berry-Esseen-Type Gumbel-Limit Theorem for Multivariate Maxima. 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. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fill:LIPIcs.AofA.2026.21,
author = {Fill, James Allen},
title = {{A New Fine-Scale Berry-Esseen-Type Gumbel-Limit Theorem for Multivariate Maxima}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {21:1--21:20},
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.2026.21},
URN = {urn:nbn:de:0030-drops-262924},
doi = {10.4230/LIPIcs.AofA.2026.21},
annote = {Keywords: Multivariate maxima, Gumbel distributions, Berry-Esseen-type theorem, Poisson approximation, Chen-Stein method, multivariate Pareto records}
}
Document
A Distributional Analysis of QuickXsort for Mergesort
Abstract
QuickXsort is an efficient in situ sequential sorting algorithm that mixes Hoare’s Quicksort algorithm with another sorting algorithm X, such as Heapsort, Insertionsort or Mergesort. The advantage is that QuickXsort can be in-place even if X is not. QuickXsort works recursively like Quicksort but uses sorting algorithm X on one of the sub-lists generated in each step. While the expected complexity of QuickXsort, measured by the number of key comparisons, has been investigated for various choices of X, here the asymptotic variance and distribution of the normalized complexity are studied with Mergesort used as X. Various versions of Mergesort and splitting regimes for the decomposition of the list by Quicksort are considered and periodicities in moments and the distributions are characterized.
Cite as
Jasper Ischebeck, Florian Lesny, and Ralph Neininger. A Distributional Analysis of QuickXsort for Mergesort. 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. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ischebeck_et_al:LIPIcs.AofA.2026.22,
author = {Ischebeck, Jasper and Lesny, Florian and Neininger, Ralph},
title = {{A Distributional Analysis of QuickXsort for Mergesort}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {22:1--22:14},
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.2026.22},
URN = {urn:nbn:de:0030-drops-262932},
doi = {10.4230/LIPIcs.AofA.2026.22},
annote = {Keywords: QuickXsort, Analysis of Algorithms, Mergesort, distribution analysis, limit law, weak convergence, contraction method}
}
Document
Moment Statistics in the Boltzmann Probability Model
Abstract
A rich family of labeled as well as unlabeled combinatorial structures is accessible by using so-called admissible specifications for which direct access to (counting) generating function equations exists. Using methods from analytic combinatorics, this quite often provides access to asymptotics for their coefficients and thus average-case statistics and knowledge on higher moments and distributions of structural parameters. Furthermore, admissible specifications are the foundation for different approaches of random sampling algorithms either uniformly for a fixed size (e.g., by unranking a random rank) or for random sizes in the Boltzmann model. The latter is of special interest for its efficiency in case of approximate size sampling. It is standard to derive asymptotics for moments from coefficients of generating functions for analytical purposes, e.g. using the saddle-point or the 𝒪-transfer method. In this paper we highlight connections between such asymptotics and the values of generating functions as computed for Boltzmann samplers. We show their use to derive fixed-size statistics from random-size Boltzmann samples. Furthermore, we introduce a new approach for the (leading term) average-case (and higher moment) analysis of structural parameters of combinatorial objects that makes the computation of generating function coefficients superfluous.
Cite as
Markus E. Nebel. Moment Statistics in the Boltzmann Probability Model. 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. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nebel:LIPIcs.AofA.2026.23,
author = {Nebel, Markus E.},
title = {{Moment Statistics in the Boltzmann Probability Model}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {23:1--23:15},
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.2026.23},
URN = {urn:nbn:de:0030-drops-262944},
doi = {10.4230/LIPIcs.AofA.2026.23},
annote = {Keywords: Boltzmann model, random sampling, average-case analysis}
}
Document
Fractional vs. Expectation Thresholds: Random Support Case
Abstract
A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. We prove for the unweighted case that this is a.a.s. true when the support is a random hypergraph.
Cite as
Thomas Fischer and Yury Person. Fractional vs. Expectation Thresholds: Random Support Case. 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. 24:1-24:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fischer_et_al:LIPIcs.AofA.2026.24,
author = {Fischer, Thomas and Person, Yury},
title = {{Fractional vs. Expectation Thresholds: Random Support Case}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {24:1--24:10},
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.2026.24},
URN = {urn:nbn:de:0030-drops-262950},
doi = {10.4230/LIPIcs.AofA.2026.24},
annote = {Keywords: Expectation Threshold, fractional Expectation Threshold, random Hypergraph}
}
Document
The Dispersion Process Has the Same Phase Transition on Almost Every Graph
Abstract
The Dispersion process was introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018), in which a number of particles are initially placed on a given vertex of a graph and update their positions according to the following dynamics. In each round, the particles which are not alone on a vertex (called unhappy) simultaneously move to a uniformly random neighbor. In contrast, the rest of the particles (called happy) stay put. The process terminates once every particle is happy.
When the process runs on the complete graph, they showed that there is a phase transition with respect to the running time when the number of particles reaches n/2. Below this threshold the running time is logarithmic, while above the threshold it is exponential. We show that the same behavior holds for the binomial random graph G(n, 1/2) with high probability.
The main difference to the complete graph is that the number of happy particles in the neighborhood of a vertex can vary significantly from vertex to vertex, resulting in differing local behaviors. Fortunately the number of such vertices is limited and thus they only have a negligible effect on the running time of the process.
Cite as
Julius Hallmann, Kostas Lakis, and Tamás Makai. The Dispersion Process Has the Same Phase Transition on Almost Every Graph. 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. 25:1-25:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hallmann_et_al:LIPIcs.AofA.2026.25,
author = {Hallmann, Julius and Lakis, Kostas and Makai, Tam\'{a}s},
title = {{The Dispersion Process Has the Same Phase Transition on Almost Every Graph}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {25:1--25:11},
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.2026.25},
URN = {urn:nbn:de:0030-drops-262960},
doi = {10.4230/LIPIcs.AofA.2026.25},
annote = {Keywords: Dispersion process, Random Graphs, Drift analysis}
}
Document
Graphical Balanced Allocations with Removals
Abstract
We study balanced allocations on graphs with removals. Load arrives at each edge e at an exponential rate and is then allocated to the vertex incident to e with the lowest current load. Load is removed from each vertex at an exponential rate. We identify a "conductance-like" quantity that determines if an equilibrium exists and allows us to bound the maximal load at equilibrium. Our analysis, based on simple potential function arguments, is very robust and can also handle noise in how the load is allocated.
We also apply our general techniques to study the synchronous version of the process above, in which allocations and removals happen simultaneously at discrete time steps. We prove that, for any regular graph, in equilibrium, the expected difference in load across an edge, averaged over all edges, is at most 2. This implies, for example, that the two-choice process on the cycle has an O(n) gap between maximal and minimal load, improving the state-of-the-art by a log n factor.
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Sam Olesker-Taylor, Thomas Sauerwald, and Luca Zanetti. Graphical Balanced Allocations with Removals. 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. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{oleskertaylor_et_al:LIPIcs.AofA.2026.26,
author = {Olesker-Taylor, Sam and Sauerwald, Thomas and Zanetti, Luca},
title = {{Graphical Balanced Allocations with Removals}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {26:1--26:16},
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.2026.26},
URN = {urn:nbn:de:0030-drops-262974},
doi = {10.4230/LIPIcs.AofA.2026.26},
annote = {Keywords: balanced allocations, load balancing, queueing networks, balls-into-bins, Markov chains}
}
Document
Asymptotics of Parking Search in Hyperfractal Networks
Abstract
We study the asymptotic behaviour of the distance to the first available parking slot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, where slot-release events occur according to Poisson processes along the streets.
We establish, by analysing the associated self-similar harmonic sums via Mellin-transform asymptotics [Flajolet et al., 1995], a power-law decay of the expected distance as the total intensity grows, with exponent equal to the inverse of the hyperfractal dimension. In particular, the scaling exponent depends only on the large-scale geometry of the network.
We further prove that this exponent is robust under random multiplicative modulations of the street intensities: mild stochastic heterogeneity affects only the multiplicative constant. Similar scaling behaviour holds for the variance, the number of turns before parking, and for a jump-over variant of the search strategy.
Cite as
Geoffrey Deperle, Christine Fricker, Philippe Jacquet, Bernard Mans, and Alessia Rigonat. Asymptotics of Parking Search in Hyperfractal Networks. 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. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deperle_et_al:LIPIcs.AofA.2026.27,
author = {Deperle, Geoffrey and Fricker, Christine and Jacquet, Philippe and Mans, Bernard and Rigonat, Alessia},
title = {{Asymptotics of Parking Search in Hyperfractal Networks}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {27:1--27:16},
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.2026.27},
URN = {urn:nbn:de:0030-drops-262982},
doi = {10.4230/LIPIcs.AofA.2026.27},
annote = {Keywords: Recursive weighted networks, Mellin transform, Asymptotic analysis, Scaling laws}
}
Document
Bounded Linear Probing Hashing
Abstract
We introduce a process that inserts elements into a hash table with a bounded number of probes, motivated by an application in cryptography. The cost of this algorithm is the number of insertion trials, whether successful or failed, until the table gets completely filled. This gives an interpolation between linear probing hashing and the coupon collector problem. We show that the process is related to a non-linear differential equation, which allows us to obtain the generating function of full tables. The proofs involve a full algebra of operators, which are themselves of independent interest. Then, we obtain the asymptotic behaviour of the expected number of insertion trials to get a full table.
Cite as
Ahmed Alharbi, Cyril Banderier, and Charles Bouillaguet. Bounded Linear Probing Hashing. 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. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alharbi_et_al:LIPIcs.AofA.2026.28,
author = {Alharbi, Ahmed and Banderier, Cyril and Bouillaguet, Charles},
title = {{Bounded Linear Probing Hashing}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {28:1--28:17},
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.2026.28},
URN = {urn:nbn:de:0030-drops-262995},
doi = {10.4230/LIPIcs.AofA.2026.28},
annote = {Keywords: Linear probing hashing, analytic combinatorics, cryptography}
}
Document
Enumeration of Bipartite Acyclic Digraphs
Abstract
We consider the asymptotic enumeration of labelled acyclic digraphs (DAGs) with the additional restriction of being bipartite. The analysis leads us to a meromorphic generating function in two variables for the number of bicoloured labelled DAGs whose analysis falls within the scope of analytic combinatorics in several variables. This allows us to obtain asymptotic formulas for the total number of labelled bipartite DAGs with a given number of vertices as well as for the number of such DAGs with a given bipartition (i.e., with prescribed sizes of the two partite sets).
Cite as
Guan-Huei Duh, Philipp Sprüssel, and Stephan Wagner. Enumeration of Bipartite Acyclic Digraphs. 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. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{duh_et_al:LIPIcs.AofA.2026.29,
author = {Duh, Guan-Huei and Spr\"{u}ssel, Philipp and Wagner, Stephan},
title = {{Enumeration of Bipartite Acyclic Digraphs}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {29:1--29:18},
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.2026.29},
URN = {urn:nbn:de:0030-drops-263005},
doi = {10.4230/LIPIcs.AofA.2026.29},
annote = {Keywords: bipartite acyclic digraph, asymptotic enumeration, analytic combinatorics in several variables}
}
Document
Asymptotic Analysis of Generating Functions Arising from Dynamic Graphs
Abstract
We study generating functions arising from sequentially growing labeled graphs where at each step either a new vertex is created or a new edge between two existing vertices is added. We provide explicit representations of the generating functions and derive asymptotic formulas for their coefficients using Laplace’s method and Bessel function approximations in the undirected model, and Hayman admissibility combined with the saddle point method in the directed model. Finally, we study a natural parameter of each model and indicate further parameter studies.
Cite as
Nadja Azzouz, Olivier Bodini, Francis Durand, and Bernhard Gittenberger. Asymptotic Analysis of Generating Functions Arising from Dynamic Graphs. 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. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azzouz_et_al:LIPIcs.AofA.2026.30,
author = {Azzouz, Nadja and Bodini, Olivier and Durand, Francis and Gittenberger, Bernhard},
title = {{Asymptotic Analysis of Generating Functions Arising from Dynamic Graphs}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {30:1--30:16},
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.2026.30},
URN = {urn:nbn:de:0030-drops-263010},
doi = {10.4230/LIPIcs.AofA.2026.30},
annote = {Keywords: combinatorial enumeration, generating functions}
}
Document
Lee-Yang Phenomena in Edge-Coloured Graph Counting
Abstract
We describe the accumulation of complex zeros of a univariate polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem from statistical physics. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral.
Cite as
Maximilian Wiesmann. Lee-Yang Phenomena in Edge-Coloured Graph Counting. 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. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wiesmann:LIPIcs.AofA.2026.31,
author = {Wiesmann, Maximilian},
title = {{Lee-Yang Phenomena in Edge-Coloured Graph Counting}},
booktitle = {37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
pages = {31:1--31:17},
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.2026.31},
URN = {urn:nbn:de:0030-drops-263028},
doi = {10.4230/LIPIcs.AofA.2026.31},
annote = {Keywords: Chromatic Polynomial, edge-coloured Graph, asymptotic Enumeration, exponential Integral, Lee-Yang Zeros, Ising Model, Partition Function}
}