S. Rinaldi

A set of well-defined operations on succession rules

In this paper we introduce a system of well-defined operations on the set of succession rules. These operations allow us to tackle combinatorial enumeration problems simply by using succession rules instead of generating functions. Finally we suggest several open problems the solution of which should lead to an algebraic characterization of the set of succession rules.

On the shape of permutomino tiles

In this paper we explore the connections between two classes of polyominoes, namely the permutominoes and the pseudo-square polyominoes. A permutomino is a polyomino uniquely determined by a pair of permutations. Permutominoes, and in particular convex permutominoes, have been considered in various kinds of problems such as: enumeration, tomographical reconstruction, and algebraic characterization. On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the plane by translation. The characterization of such objects has been given by Beauquier and Nivat, who proved that a polyomino tiles the plane by translation if and only if it is a pseudo-square or a pseudo-hexagon. In particular, a polyomino is pseudo-square if its boundary word may be factorized as [emailxa0protected]^[emailxa0protected]^, where [emailxa0protected]^ denotes the path X traveled in the opposite direction. In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that [emailxa0protected]^[emailxa0protected]^ is pseudo-square is prefix of a unique infinite word Y~ with period 4|X|N|X|E. Also, we show that [emailxa0protected]^[emailxa0protected]^ are centrosymmetric, i.e. they are fixed by rotation of angle @p. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods.

Emission Discrete Tomography

Three problems of emission discrete tomography (EDT) are presented. The first problem is the reconstruction of measurable plane sets from two absorbed projections. It is shown that Lorentz theorems can be generalized to this case. The second is the reconstruction of binary matrices from their absorbed row and columns sums if the absorption coefficient is μ0 = log((1+v/5)/2). It is proved that the reconstruction in this case can be done in polynomial time. Finally, a possible application of EDT in single photon emission computed tomography (SPECT) is presented: Dynamic structures are reconstructed after factor analysis.