Vector Partition Functions

The problem of enumerating vector partitions is the d-dimensional analogue of the well-studied coin exchange problem. Given a set of vectors a1, . . . , an ∈ Z d , the vector partition function yields the number of solutions to a1x1 + . . . , anxn = b as a function of b. One can view this as the enumeration of integer points in the polytope {x ∈ N n : Ax = b, x ≥ 0} where A is the matrix whose columns are a1, . . . , an. The vector partition function pA associated to the matrix A takes b as input and returns the corresponding number of vector partitions. Sturmfels (1994) showed that vector partition function can be represented explicitly as a piecewise quasi-polynomial (roughly a polynomial with periodic coefficients) whose domains of quasi-polynomiality are the maximal cones (chambers) of a fan (called the chamber complex) associated to the matrix A. In addition, Sturmfels and De Loera (2003) showed that if A is unimodular (every square submatrix has determinant 0, ±1), then the quasi-polynomials are actually each polynomials. We show that for certain chambers of A (which we call external chambers) the associated quasi-polynomial arises from a coin exchange problem, and is univariate after an appropriate change of variables. Additionally, we show that if A is unimodular, then the polynomial associated to an external chamber is given by a negative binomial coefficient which depends on a single facet of the chamber. We also show that one can easily calculate linear factors of polynomials associated to other chambers of A (which we call semi-external chambers) in the case that A is unimodular. The Littlewood-Richardson and Kronecker coefficients are two different sets of structure constants associated to the Schur polynomials. Rassart (2004) and Mishna, Rosas, Sundaram (2021) have considered vector partition function approaches to computing LittlewoodRichardson and Kronecker coefficients respectively. We exploit Rassart's approach in order to derive a new determinantal formula for the Littlewood-Richardson coefficients associated to GL3. We also use it to give a novel geometrical interpretation of a well-known stability result. Additionally, we address some answers related to symmetries of the Littlewood-Richardson coefficients, partially by computing the chamber complex for the Littlewood-Richardson coefficients associated to GL4. In our work on Kronecker coefficients, we use the vector partition function approach to create a computational tool for Kronecker coefficients with partition lengths bounded by 2, 4, and 8. Additionally, we obtain vanishing conditions and generate a stable face of the Kronecker polyhedron. Finally, we obtain new upper bounds for the Kronecker coefficients, which in some cases seem to be the best known.