Partitions and Compositions Defined by Inequalities

Abstract

We consider sequences of integers (λ₁,..., λ _k ) defined by a system of linear inequalities λ _i ≥ ∑ _j>ia_ijλ _j with integer coefficients. We show that when the constraints are strong enough to guarantee that all λ _i are nonnegative, the generating function for the integer solutions of weight n has a finite product form $\prod_{i} (1-q^{b_i})^{-1}$, where the b _i are positive integers that can be computed from the coefficients of the inequalities. The results are proved bijectively and are used to give several examples of interesting identities for integer partitions and compositions. The method can be adapted to accommodate equalities along with inequalities and can be used to obtain multivariate forms of the generating function. We show how to extend the technique to obtain the generating function when the coefficients a_i,i+1 are allowed to be rational, generalizing the case of lecture hall partitions. Our initial results were conjectured thanks to the Omega package (G.E. Andrews, P. Paule, and A. Riese, European J. Comb. 22(7) (2001), 887–904).