Restricted partitions

We prove a known partitions theorem by Bell in an elementary and constructive way. Our proof yields a simple recursive method to compute the corresponding Sylvester polynomials for the partition. The previous known methods to obtain these polynomials are in general not elementary. 1. Proof of Bell's theorem. The main purpose of this section is to prove the following theorem, originally proved by Bell in [1], by elementary methods. Theorem 1.1. For a fixed positive integer n, let A1,...,An be positive integers and let M be their least common multiple. For a fixed integer r , the number of nonnegative solutions Xn,...,X1 of An·Xn+···+A1·X1 = M K +r , which we indicate by D n (M K + r), is given by a polynomial in K, which is either the zero polynomial or a polynomial with rational coefficients of degree n − 1. First, we need the following known result. Lemma 1.2. For N ≥ 0 and m ≥ 1, H m (N) = 0 m + 1 m +···+N m is a polynomial in N of degree m + 1 with rational coefficients. Besides, H m (−1) = 0. For example, we have H 1 (N) = 1 2 N 2 + 1 2 N, H 2 (N) = 1 3 N 3 + 1 2 N 2 + 1 6 N. There exist several elementary methods to obtain the polynomials H m (N). We will see that D n (M K +r) is a polynomial as a direct consequence of Lemma 1.2. The proof of Theorem 1.1. We are going to prove Bell's theorem by mathematical induction. The theorem is clearly true for n = 1 since in this case, the number of solutions to the equation A1 · X1 = A1 · K + r ' is given by the polynomials D 1 (A1 · K + r) = 1 if r is multiple of A1, D 1 (A1 · K + r) = 0 if r is not a multiple of A1. (1.2) Let n ≥ 1 be a given, and assume Theorem 1.1 holds for n − 1; we will prove it is also true for n.