A Reciprocity Law Arising From Lisonek Quasi-Polynomials To Count Isomorphism Types Of Block Designs

Ehrhart proved in 1962 that the number of lattice-points in the integral dilates of a rational convex polytope can be counted by a quasi-polynomial depending on the dimension of the polytope. Beck extended his idea to semi-dilations of rational polytopes. In this case, one can count the lattice-points via a multivariate quasi-polynomial. Recently, Lisonek introduced a more general definition of quasi-polynomial in order to count isomorphism types of block designs, dissections of regular polygons, linear codes and unrestricted codes.

On the other hand, throughout combinatorics, many counting functions that are quasi-polynomials yield combinatorial information by evaluating them at negative integers. If this is the case, we say that we have a reciprocity law. For example, Ehrhart and Macdonald proved a reciprocity law relating the Ehrhart quasi-polynomial of a polytope with the lattice-points in the interior of the polytope.

One of the main goals of our research is to derive reciprocity laws arising from applying Lisonek quasi-polynomials to certain combinatorial families. Specifically, for the problems we are considering, one can associate a polytope to these combinatorial families. Then the problem of counting isomorphism types is translated to counting lattice-points in this polytope. Therefore, Ehrhart-Macdonald reciprocity applies. However, we want to know what exactly these lattice-points in the interior of the polytope correspond to in the combinatorial family. In other words, we want to determine what kind of combinatorial objects the Lisonek quasi-polynomial counts when evaluated at negative integers.