Invited Talks

Emilie Charlier - "Generating expansions in Cantor real bases via a transducer".

Representing real numbers using suitable numeration systems — such as integer bases, beta-expansions, or Cantor bases — has long been a central topic in mathematics. In this talk, I will focus on Cantor real bases, and more specifically on automatic Cantor real bases and the properties of real number expansions in this framework.

I will introduce a single transducer associated with a fixed real number r, which computes the B-expansion of r for an infinite family of Cantor real bases B given as input. This point of view contrasts with traditional computational models, where the numeration system is fixed in advance. Under suitable assumptions on the finitely many Pisot numbers occurring in the Cantor real base, only a finite portion of the transducer is visited. I will then present several fundamental results concerning both the structure of this transducer and decidability questions related to these expansions. In particular, for certain classes of Cantor real bases, important combinatorial properties — such as greediness or periodicity of the expansion — can be decided algorithmically. This talk is based on joint work with Pierre Popoli and Michel Rigo.

Sergiey Kirgizov - "Q-bonacci words and related structures".

The Q-bonacci (also known as q-decreasing) words are binary, and all their locally maximal factors of the form 0ᵃ1ᵇ satisfy qa > b or a = 0. We explore the formation and enumeration of such words in three cases of q: natural, rational and real. When q is natural, such words are enumerated by the classical (q+1)-bonacci sequence (next term is the sum of q+1 previous terms). When we go from natural to rational and real cases, the situation becomes more and more mesmerizing. Many connections with recent, classical well-known, as well as fairly esoteric mathematical structures will be discussed during this talk.

Simone Rinaldi - "Ascending polyominoes"

Convex polyominoes can be refined according to the number of direction

changes in monotone paths connecting pairs of cells, leading to the

notion of k-convexity. In particular, the cases k=1 and k=2 correspond to L-convex and Z-convex polyominoes, two classical subclasses exhibiting very different combinatorial behaviours.

Between these two extremal families lie several intermediate classes,

such as centered and 4-stack polyominoes, which form a natural

hierarchy, L-convex ⊆ centered ⊆ 4-stacks ⊆ Z-convex.

A striking feature of this hierarchy is the transition in the nature

of the generating functions and asymptotic growth: while L-convex

and centered polyominoes have rational generating functions and

exponential growth of order (2+\sqrt{2})^n and 4^n, respectively,

the classes of 4-stack, Z-convex, and general convex polyominoes

are described by algebraic generating functions with asymptotics

of order \sqrt{n}4^n or n4^n.

In this work we focus on Z-convex polyominoes and introduce a finer

structural analysis based on the notions of NW- and NE-convexity degree.

This refinement yields a decomposition into three disjoint subclasses,

denoted by C(1,2), C(2,1), and C(2,2), providing a more detailed understanding of the internal structure of Z-convex polyominoes.

To enumerate these families, we introduce the class of ascending polyominoes, characterized by a simple geometric condition.

We show that convex polyominoes can be decomposed as

Convex = Ascending + Descending + C(2,2),

where ascending and descending polyominoes are equinumerous and their intersection coincides with L-convex polyominoes.

The enumeration is carried out via the application of the ECO method and a generating tree construction, leading to functional equations for the corresponding generating functions. By solving these equations, we obtain explicit algebraic expressions for the generating functions of ascending polyominoes, centered ascending polyominoes and

of the subclasses C(1,2), C(2,1), and C(2,2).

Finally, we derive the asymptotic growth of these families, showing that

C(1,2) and C(2,1) have growth of order n4^n, while the class

C(2,2) exhibits growth of order \sqrt{n}4^n, revealing a precise combinatorial transition within the hierarchy of convex polyominoes.