External columns and chambers of vector partition functions

TL;DR

This paper introduces external columns and chambers in vector partition functions, linking certain chambers to coin exchange problems and deriving formulas and conditions for polynomiality, with applications to graph enumeration.

Contribution

It defines external chambers and shows their associated quasi-polynomials relate to lower-dimensional vector partition functions, including a determinantal formula and polynomiality criteria.

Findings

External chambers correspond to coin exchange problems.

Quasi-polynomials in external chambers are given by negative binomial coefficients.

Results apply to enumerating loopless multigraphs with degree constraints.

Abstract

The vector partition function $p_A$ associated to a $d \times n$ matrix $A$ with integer entries is the function $\mathbb{Z}^d \to \mathbb{N}$ defined by $\mathbf{b} \to \#\{\mathbf{x} \in \mathbb{N}^n : A\mathbf{x} = \mathbf{b}\}$. It is known that vector partition functions are piecewise quasi-polynomials whose domains of quasi-polynomiality are maximal cones (chambers) of a fan called the chamber complex of $A$. In this article we introduce \emph{external columns} and \emph{external chambers} of vector partition functions. Our main result is that (up to a saturation condition) the quasi-polynomial associated to a chamber containing external columns arises from a vector partition function with $k$ fewer equations and variables. In the case that the chamber is external -- that is, when the number of external columns in a chamber is as large as possible without being trivial -- the quasi-polynomial arises from a coin exchange problem. By exploiting this we are able to obtain a determinantal formula, characterize when the quasi-polynomial is polynomial, and show that in this case it is actually given by a negative binomial coefficient. We then apply these results to the enumeration of loopless multigraphs satisfying some degree conditions. Finally, we suggest a generalization to a result of Baldoni and Vergne for polynomials arising from chambers that we call \emph{semi-external chambers}.