This section seeks high quality research articles in all aspects of combinatorics, including enumerative combinatorics, probabilistic combinatorics, extremal combinatorics, algebraic combinatorics, additive combinatorics, bijections and mappings to enumeration, structural and enumerative properties of combinatorial objects, ordered sets, posets, quasi-orderings, combinatorial structures with geometric properties, combinatorial geometry, combinatorial objects in statistical physics, positional games, power series and generating functions.
Editors:Guillaume Chapuy, Sergi Elizalde, Stefan Felsner, Anna Frid, David Galvin, Mihyun Kang, Martin Klazar, Caroline Klivans, Kolja Knauer, Christian Krattenthaler, Daniel Krob, Piotr Micek, Jean-Christophe Novelli, Jay Pantone, Milos Stojakovic
Editors:Guillaume Chapuy, Sergi Elizalde, Stefan Felsner, Anna Frid, David Galvin, Mihyun Kang, Martin Klazar, Caroline Klivans, Kolja Knauer, Christian Krattenthaler, Daniel Krob, Piotr Micek, Jean-Christophe Novelli, Jay Pantone, Milos Stojakovic
Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?
Alexander Dobler, Stephen Kobourov, Debajyoti Mondal, Martin Nöllenburg
Abstract
We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a) restricting the curves to be lines or line segments, (b) allowing two curves to cross if they do not share an element, or not; and (c) allowing two curves to overlap or not. We show $\exists\mathbb{R}$-hardness for six of the eight resulting decision problem variants and describe polynomial-time algorithms in some restricted settings. Lastly, we briefly touch on what happens if we allow the lines of the represented hyperedges to have bends - to this we generalize a counterexample to a long-standing result that was sometimes assumed to be correct. 24 pages, 12 figures
Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order $\sqrt n$. Suggestions by two reviewers are included. Third version contains some formatting
K. Adaricheva, A. Mata, S. Silberger, A. Zamojska-Dzienio
Abstract
We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices. 21 pages, 11 figures
In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\mathtt{area}$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\mathtt{depth}$ to so-called ``$\vec{k}$-Dyck paths''. They showed that $\mathtt{area}$ and $\mathtt{depth}$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of Łukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent. 8 pages, 4 figures, accepted by Discrete Mathematics & Theoretical Computer Science, final version
Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational. Finally, we also examine the analogous question for the Tribonacci word and the Thue-Morse word.
The symmetric random walk is known to be recurrent in one and two dimensions, and becomes transient in three or higher dimensions. We compare the symmetric random walk to walks driven by certain \polya\ urns. We show that, in contrast, if the probabilities of the random walk are instead driven by a \polya-Eggenberger urn, the states are recurrent only in one dimension. Further consideration of exchangeability reveals that the walk is null recurrent. As soon as the underlying Markov chain of \polya\ walk gets in two dimensions or higher, there is a positive probability that the walker gets lost in the space, and the probability of her recurrence is less than 1. On the other hand, a walk driven by Friedman urn behaves like the symmetric random walk, being recurrent in one and two dimensions and transient in higher dimensions. As Friedman urn scheme is not exchangeable, it is considerably harder to determine the nature of the recurrence in one and two dimensions. Empirical evidence through simulation suggests that in one dimension Friedman walk is positive recurrent.
Lubomíra Dvořáková, Edita Pelantová, Jeffrey Shallit
Abstract
In 2017, Clark Kimberling defined an interesting sequence ${\bf B} = 0100101100 \cdots$ of $0$'s and $1$'s by certain inflation rules, and he made a number of conjectures about this sequence and some related ones. In this note we prove his conjectures using, in part, the Walnut theorem-prover. We show how his word is related to the infinite Tribonacci word, and we determine both the subword complexity and critical exponent of $\bf B$.
Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, pro_K, on m-packed labelings of tableaux. The operator pro was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of pro_K on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of pro_K exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree. Made corrections asked for by the referee
It is known that any two trees on the same $n$ leaves can be displayed by a network with $n-2$ reticulations, and there are two trees that cannot be displayed by a network with fewer reticulations. But how many reticulations are needed to display multiple trees? For any set of $t$ trees on $n$ leaves, there is a trivial network with $(t - 1)n$ reticulations that displays them. To do better, we have to exploit common structure of the trees to embed non-trivial subtrees of different trees into the same part of the network. In this paper, we show that for $t \in o(\sqrt{\lg n})$, there is a set of $t$ trees with virtually no common structure that could be exploited. More precisely, we show for any $t\in o(\sqrt{\lg n})$, there are $t$ trees such that any network displaying them has $(t-1)n - o(n)$ reticulations. For $t \in o(\lg n)$, we obtain a slightly weaker bound. We also prove that already for $t = c\lg n$, for any constant $c > 0$, there is a set of $t$ trees that cannot be displayed by a network with $o(n \lg n)$ reticulations, matching up to constant factors the known upper bound of $O(n \lg n)$ reticulations sufficient to display \emph{all} trees with $n$ leaves. These results are based on simple counting arguments and extend to unrooted networks and trees.
We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution towards favoring permutations with more or fewer fixed points than is typical under the uniform distribution. One case we study features a phase transition where the limiting distribution changes abruptly from negative binomial to Rayleigh to normal depending on the bias parameter. 17 pages, final version
Probabilistic counters are well-known tools often used for space-efficient set cardinality estimation. In this paper, we investigate probabilistic counters from the perspective of preserving privacy. We use the standard, rigid differential privacy notion. The intuition is that the probabilistic counters do not reveal too much information about individuals but provide only general information about the population. Therefore, they can be used safely without violating the privacy of individuals. However, it turned out, that providing a precise, formal analysis of the privacy parameters of probabilistic counters is surprisingly difficult and needs advanced techniques and a very careful approach. We demonstrate that probabilistic counters can be used as a privacy protection mechanism without extra randomization. Namely, the inherent randomization from the protocol is sufficient for protecting privacy, even if the probabilistic counter is used multiple times. In particular, we present a specific privacy-preserving data aggregation protocol based on Morris Counter and MaxGeo Counter. Some of the presented results are devoted to counters that have not been investigated so far from the perspective of privacy protection. Another part is an improvement of previous results. We show how our results can be used to perform distributed surveys and compare the properties of counter-based solutions and a standard Laplace method.
We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between random Pólya trees and random unlabelled trees tends to zero when the number of vertices tends to infinity. In order to demonstrate that our approach is not restricted to trees we extend our results to tree-like classes of graphs.
Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed $k$-ary tree where we place $k^n$ chips labeled $0,1,\dots, k^n-1$ on the root for some nonnegative integer $n$, and we say a vertex $v$ can fire if it has at least $k$ chips. When a vertex fires, we select $k$ labeled chips and send the $i$th smallest chip among them to its $i$th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of $1, 2, \dots, n$. We then express the stable configuration as a permutation of $0,1, 2, \dots, k^n-1$ and explore its properties, such as the number of inversions and descents. 20 pages, 4 figures, 1 table, v4: final version
In this paper, we study a class of cellular automata (CA) called stable cellular automata (SCA) that preserve stability by reflection, modulo-recurrent, and richness. After applying these automata to Sturmian words, we determine some of their combinatorial properties. Next, we calculate the classical and palindromic complexity functions of these words. Finally, we demonstrate that these words are $2$-balanced and establish their abelian complexity function. 17 pages, 0 figure, to be published in Discrete Mathematics and Theoretical Computer Science (DMTCS)
This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. Many maps of interest in dynamical algebraic combinatorics, such as rowmotion of order ideals, can be represented as a composition of "toggling" involutions, each of which modifies its object only locally. Similarly whirling is made up of locally-acting whirling maps which directly generalize toggles, but cycle through more than two possible outputs. In this first paper on whirling, we consider it as a map on subfamilies of functions between finite sets. For whirling acting on the set of injections or the set of surjections, we prove that within each whirling orbit, any two elements of the codomain appear as outputs of functions the same number of times. This result can be stated in terms of the homomesy phenomenon, which occurs when a statistic has the same average across every orbit. We further explore homomesy results and conjectures for whirling on restricted-growth words, which correspond to set partitions. These results extend the collection of combinatorial objects for which we have interesting dynamics and homomesy, and open the door to considering whirling in other contexts. 21 pages, 5 figures
We introduce a parametrization of the conjugates of Christoffel words based on the integer Ostrowski numeration system. We use it to give a precise description of the borders (prefixes which are also suffixes) of the conjugates of Christoffel words and to revisit the notion of Sturmian graph introduced by Epifanio et al. 32 pages
Carla Groenland, Tom Johnston, Jamie Radcliffe, Alex Scott
Abstract
We investigate the following generalisation of permutation networks. We say a sequence $T=(T_1,\dots,T_\ell)$ of transpositions in $S_n$ forms a $t$-reachability network if, for every choice of $t$ distinct points $x_1, \dots, x_t\in \{1,\dots,n\}$, there is a subsequence of $T$ whose composition maps $j$ to $x_j$ for every $1\leq j\leq t$. When $t=n$, any permutation in $S_n$ can be created and $T$ is a permutation network. Waksman [JACM, 1968] showed that the shortest permutation networks have length about $n \log_2(n)$. In this paper, we investigate the shortest $t$-reachability networks for other values of $t$. Our main result settles the case of $t=2$: the shortest $2$-reachability network has length $\lceil 3n/2\rceil-2 $. For fixed $t \geq 3$, we give a simple randomised construction which shows that there exist $t$-reachability networks with $(2+o_t(1))n$ transpositions. We also study the effect of restricting to star-transpositions, i.e. restricting all transpositions to have the form $(1, \cdot)$. 12 pages, 1 figure
In the $(a,b)$-biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph $G$ such that Dominator selects $a$ and Staller selects $b$ vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of $G$, while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer $b$, Dominator's threshold, $\textrm{a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, $\textrm{a}'_b$ denotes the minimum $a$ such that Dominator wins when Staller starts the $(a,b)$-biased game. Staller's thresholds, $\textrm{b}_a$ and $\textrm{b}'_a$, are defined analogously. It is proved that Staller wins the $(k-1,k)$-biased games in a graph $G$ if its order is sufficiently large with respect to a function of $k$ and the maximum degree of $G$. Along the way, the $\ell$-local domination number of a graph is introduced. This new parameter is proved to bound Dominator's thresholds $\textrm{a}_\ell$ and $\textrm{a}_\ell'$ from above. As a consequence, $\textrm{a}_1'(G)\le 2$ holds for every claw-free graph $G$. More specific results are obtained for thresholds in line graphs and Cartesian grids. Based on the concept of $[1,k]$-factor of a graph $G$, we introduce the star partition width $σ(G)$ of $G$, and prove that $\textrm{a}_1'(G)\le σ(G)$ holds for any nontrivial graph $G$, while $\textrm{a}_1'(G)=σ(G)$ if $G$ is a tree.
, Stefan Felsner