Restricted partitions (Q1777652)

From the text: ``We prove a known partitions theorem by \textit{E. T. Bell} [Am. J. Math. 65, 382--386 (1943; Zbl 0060.09603)] 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. Theorem: Let \(A_1,A_2,\dots,A_n\) be positive integers with \(M'\) their least common multiple. For a fixed integer \(r'\), the number of non-negative solutions of \[ A_1x_1 + A_2x_2 + \dots + A_nx_n = M'K + r' \] is given by a polynomial in \(K\), which is either \(0\) or a polynomial with rational coefficients of degree \(n-1\).'' As an example, the author computes the polynomial \(D_3(12K+8)\), that is, for the equation \(4x_3 + 3x_2 + 2x_1 = 12K + 8\), this polynomial is \(3^2 + 7K + 4\). This is also done for \(D_2(6K)\), \(D_2(6K+4)\), and \(D_2(6K+8)\). The printing is marred by missing sub- and superscripts. (Revised version)