Simone Rinaldi

An algebraic characterization of the set of succession rules

“Qui dedit beneficium taceat; narret qui accepit” (Seneca) Merci Maurice In this paper we will give a formal description of succession rules in terms of linear operators satisfying certain conditions. This representation allows us to introduce a system of well-defined operations into the set of succession rules and then to tackle problems of combinatorial enumeration simply by using operators instead of generating functions. Finally, we will suggest several open problems whose solution should lead to an algebraic characterization of the set of succession rules.

A tomographical characterization of l-convex polyominoes

Our main purpose is to characterize the class of L-convex polyominoes introduced in [3] by means of their horizontal and vertical projections. The achieved results allow an answer to one of the most relevant questions in tomography i.e. the uniqueness of discrete sets, with respect to their horizontal and vertical projections. In this paper, by giving a characterization of L-convex polyominoes, we investigate the connection between uniqueness property and unimodality of vectors of horizontal and vertical projections. In the last section we consider the continuum environment; we extend the definition of L-convex set, and we obtain some results analogous to those for the discrete case.

Jumping succession rules and their generating functions

We study a generalization of the concept of succession rule, called jumping succession rule, where each label is allowed to produce its sons at different levels, according to the production of a fixed succession rule. By means of suitable linear algebraic methods, we obtain simple closed forms for the numerical sequences determined by such rules and give applications concerning classical combinatorial structures. Some open problems are proposed at the end of the paper.

Enumeration of L-convex polyominoes by rows and columns

In this paper, we consider the class of L-convex polyominoes, i.e. the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once.Using the ECO method, we prove that the number fn of L-convex polyominoes with perimeter 2(n + 2) satisfies the rational recurrence relation fn = 4fn-1 - 2fn-2, with f0 = 1, f1 = 2, f2 = 7. Moreover, we give a combinatorial interpretation of this statement. In the last section, we present some open problems.

Reconstructing words from a fixed palindromic length sequence

To every word ω is associated a sequence Gω built by computing at each position i the length of its longest palindromic suffix. This sequence is then used to compute the palindromic defect of a finite word Ω defined by D(Ω) = |Ω|+1−|Pal(Ω)| where Pal(Ω) is the set of its palindromic factors. In this paper we exhibit some properties of this sequence and introduce the problem of reconstructing a word from GΩ. In particular we show that up to a relabelling the solution is unique for 2‐letter alphabets.

ECO method and the exhaustive generation of convex polyominoes

ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In this paper we use the ECO method and the concept of succession rule to develop an algorithm for the exhaustive generation of convex polyominoes. Then we prove that this algorithm runs in constant amortized time.

On the equivalence problem for succession rules

The notion of succession rule (system for short) provides a powerful tool for the enumeration of many classes of combinatorial objects. Often, different systems exist for a given class of combinatorial objects, and a number of problems arise naturally. An important one is the equivalence problem between two different systems. In this paper, we show how to solve this problem in the case of systems having a particular form. More precisely, using a bijective proof, we show that the classical system defining the sequence of Catalan numbers is equivalent to a system obtained by linear combinations of labels of the first one.

Some linear recurrences and their combinatorial interpretation by means of regular languages

Abstract In this paper we apply ECO method and the concept of numeration systems to give a combinatorial interpretation to linear recurrences of the kind a n = ka n−1 +ha n−2 , where k>|h|⩾0 . In particular, we define a language L such that the words of L having length n satisfy the recurrence, and then we describe a recursive construction for this language, according to the ECO method, and the corresponding finite succession rule.

Approximating algebraic functions by means of rational ones

In this paper, we use ECO method and the concept of succession rule to enumerate restricted classes of combinatorial objects. Let &O be the succession rule describing a construction of a combinatorial objects class, then the construction of the restricted class is described by means of an approximating succession rule ORk obtained from &O in a natural way. We give sufficient conditions for the rule k to be finite; finally we determine finite approximating rules for various classes of paths, and the approximation of the corresponding algebraic language with a regular one.

A tiling system for the class of L-convex polyominoes

A polyomino is said to be L-convex if any two of its cells can be connected by a path entirely contained in the polyomino, and having at most one change of direction. In this paper, answering a problem posed by Castiglione and Vaglica [6], we prove that the class of L-convex polyominoes is tiling recognizable. To reach this goal, first we express the L-convexity constraint in terms of a set of independent properties, then we show that each class of convex polyominoes having one of these properties is tiling recognizable.