The Polynomial Part of a Restricted Partition Function Related to the Frobenius Problem

Given a set of positive integers A = {a1, . . . , an}, we study the number pA(t) of nonnegative integer solutions (m1, . . . ,mn) to ∑n j=1mjaj = t. We derive an explicit formula for the polynomial part of pA. Let A = {a1, . . . , an} be a set of positive integers with gcd(a1, . . . , an) = 1. The classical Frobenius problem asks for the largest integer t (the Frobenius number) such that m1a1 + · · ·+mnan = t has no solution in nonnegative integers m1, . . . ,mn. For n = 2, the Frobenius number is (a1 − 1)(a2 − 1) − 1, as is well known, but the problem is extremely difficult for n > 2. (For surveys of the Frobenius problem, see [R, Se].) One approach [BDR, I, K, SÖ] is to study the restricted partition function pA(t), the number of nonnegative integer solutions (m1, . . . ,mn) to ∑n j=1 mjaj = t, where t is a nonnegative integer. The Frobenius number is the largest integral zero of pA(t). Note that, in contrast to the Frobenius problem, in the definition of pA we do not require a1, . . . , an to be relatively prime. In the following, a1, . . . , an are arbitrary positive integers. ∗Research partially supported by NSF grant DMS-9972648. the electronic journal of combinatorics 8 (2001), #N7 1 It is clear that pA(t) is the coefficient of z t in the generating function G(z) = 1 (1− za1) · · · (1− zan) . If we expand G(z) by partial fractions, we see that pA(t) can be written in the form ∑ λ PA,λ(t)λ , where the sum is over all complex numbers λ such that λi = 1 for some i, and PA,λ(t) is a polynomial in t. The aim of this paper is to give an explicit formula for PA,1(t), which we denote by PA(t) and call the polynomial part of pA(t). It is easy to see that PA(t) is a polynomial of degree n − 1. (More generally, the degree of PA,λ(t) is one less than the number of values of i for which λi = 1.) It is well known [PS, Problem 27] that pA(t) = tn−1 (n− 1)! a1 · · · an +O ( tn−2 ) . Our theorem is a refinement of this statement. We note that Israilov derived a more complicated formula for PA(t) in [I]. Let us define QA(t) by pA(t) = PA(t) + QA(t). From the partial fraction expansion above, it is clear that QA (and hence also pA) is a quasi-polynomial, that is, an expression of the form cd(t)t d + · · ·+ c1(t)t+ c0(t), where c0, . . . , cd are periodic functions in t. (See, for example, Stanley [St, Section 4.4], for more information about quasi-polynomials.) In the special case in which the ai are pairwise relatively prime, each PA,λ(t) for λ 6= 1 is a constant, and thus QA(t) is a periodic function with average value 0, and this property determines QA(t), and thus PA(t). Discussions of QA(t) can be found, for example, in [BDR, I, K]. We define the Bernoulli numbers Bj by z ez − 1 = ∑