On some counting problems for semi-linear sets

Let X be a subset of N or Z. We can associate with X a function GX : N t −→ N which returns, for every (n1, . . . , nt) ∈ N , the number GX(n1, . . . , nt) of all vectors x ∈ X such that, for every i = 1, . . . , t, |xi| ≤ ni. This function is called the growth function of X. The main result of this paper is that the growth function of a semi-linear set of N or Z is a box spline. By using this result and some theorems on semi-linear sets, we give a new proof of combinatorial flavour of a well-known theorem by ∗This work was partially supported by MIUR project “Aspetti matematici e applicazioni emergenti degli automi e dei linguaggi formali”. The first author acknowlegdes the partial support of fundings “Facoltà di Scienze MM. FF. NN. 2007” of the University of Rome “La Sapienza”.