Published: 24th June 2024 DOI: 10.4204/EPTCS.403 ISSN: 2075-2180 |
This volume contains the proceedings of the 13th edition of the conference GASCom, held at LABRI, Bordeaux, during the week of June 24-28, 2024.
The very first edition of GASCom took place in Bordeaux on January 18 and 19, 1994, organized by Dominique Gouyou-Beauchamps and the late Jean-Guy Penaud, in conjunction with "Polyominoes and Tilings", organized by Philippe Aigrin in Toulouse on January 20 and 21, 1994. As reported in the foreword of the dedicated special issue,
"Both meetings investigated the same types of combinatorial
objects (polynominoes, paths, animals) but with different
motivations. Counting and enumeration were among the interests common
to both, but GASCom focused on random generation, Polyominoes and
Tilings on the geometrical and topological problems in tiling and the
complexity of tiling algorithms."
(Theoretical Computer Science, 159 (1996) 1--2)
The second GASCom was organized by Jean-Guy-Penaud and Renzo Pinzani in Caen, again in conjunction with "Polyominoes and Tilings", organized by Jacques Mazoyer, under the name "Journées d'hiver à Caen", on January 30 and February 1, 1997. Again from the foreword, we quote
"They follow on from a previous edition, split between Bordeaux
and Toulouse in January 1994, and the size of the audience common to
both conferences prompted the organizers to repeat the experience.
Readers of these proceedings will appreciate the wisdom of this
choice"
(translated from French, Theoretical Computer Science, 218 vol. 2
(1999) 217--218)
In 2024, GASCom returns to its origins, since it was the Ecole de Combinatoire de Bordeaux that planted the seeds of these joint conferences, reminding us that the link between the two subjects (combinatorial generation and polyominoes) is as strong as ever.
The conference features 5 invited talks and 33 contributed talks carefully selected by the members of the program committee. We warmly thank them for their efforts and dedication in the reviewing process. We also extend our thanks to the members of the organizing committee, who have worked hard to organize the conference in a friendly and stimulating atmosphere. Finally, we thank all the speakers, whose contributions are of course the key ingredients for the success of the conference.
To conclude, we wish to express our sincere gratitude to the staff of EPTCS, whose help and collaboration have been invaluable in the preparation of this volume.
Wang tiles are squares with coloured edges that can be placed side by side as long as two neighbouring tiles have the same colour on their common side. Given a finite set of such tiles, whether it is possible to create an infinite tiling of the plane is an undecidable problem, known as the domino problem. This problem has also been studied for about fifteen years for groups of finite type other than Z^{2}. In this talk I will focus on a variant of this problem, the snake domino tiling. Instead of looking for a tiling of the whole plane (or the whole group), we only look for a tiling of an infinite path. This problem remains undecidable in the Euclidean plane Z^{2}. I will present recent progress on the snake tiling problem for groups, and make the connection with self-avoiding paths.
For a given finite Markov chain with uniform stationary distribution, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterized much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.
In 1987 Bak, Tang and Wiesenfeld introduced their famous sandpile model as a first system showing self-organized criticality. In 1988 Macdonald introduced his famous symmetric polynomials. Each of these two discoveries produced a huge amount of research that is still developing intensely today. But until recently, these two lines of research went on without any relevant interaction. In this talk we show how the combinatorics generated by these two important mathematical objects come together in a surprising way, proving that a synergy between these two topics is inevitable.
Polyforms - shapes constructed by gluing together copies of cells in an underlying grid - are a convenient experimental tool with which to probe problems in tiling theory. They are expressive: in practice, even simple families like polyominoes exhibit many of the tiling-theoretic behaviours we might wish to study. But unlike the general world of shapes, polyforms can be enumerated exhaustively and the behaviour of each one examined using discrete computation. In this way polyforms can provide the raw data from which we might identify patterns, examples, or counterexamples that drive new insights into unsolved problems in tiling theory. I am particularly interested in using polyforms to explore two open problems. The first asks whether there is a finite upper bound to a shape's Heesch number, the maximum number of times a non-tiler may be surrounded by rings of copies of itself and yet still fail to tile the plane. The second asks whether there is a finite upper bound to a shape's isohedral number, the minimum number of transitivity classes needed in any periodic tiling by the shape. I discuss these two problems and the progress made in exploring them using discrete computation. I also discuss the connections from Heesch numbers and isohedral numbers to the question of the existence of an aperiodic monotile, a problem that remained open for over sixty years but that was resolved in 2023 by a polyform called the "hat".
We will survey several results concerning random finite state automata, including random generation and algorithm analysis. We will place special focus on the subset construction, the standard algorithm for building a deterministic automaton equivalent to a given non-deterministic one.