Felix Breuer's Blog

Enumerating Colorings, Tensions and Flows in Cell Complexes

My latest math paper has hit the arXiv. It is called Enumerating Colorings, Tensions and Flows in Cell Complexes and it is joint work with Matthias Beck, Jeremy Martin and Logan Godkin. It has been great working with all of them. I am particularly pleased that this is my first joint paper with Matthias, the great guy who brought me to San Francisco!

This paper is a continuation of the work that Matthias Beck and Thomas Zaslavsky started on using inside-out polytopes to prove reciprocity results for counting polynomials and that Raman Sanyal and I adapted to the case of modular counting functions. In the present work we generalize all of this considerably, by proving a whole collection of combinatorial reciprocity theorems for flow, tension and chromatic quasipolynomials defined on cell complexes, i.e., in terms of arbitrary integer matrices. The fact that the Ehrhart theory methods developed for the graph case do generalize to cell complexes is a testament to the remarkable power of the geometric approach to combinatorics. Head over here to hear the full story!