On the power-series expansion of a rational function
Volume 62 / 1992
Abstract
Introduction. The problem of determining the formula for $P_S(n)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_{s₁},..., h_{s_k}$, of the equation h_{s₁} s₁ + ... + h_{s_k} s_k = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of$xⁿ in [(1-x^{s₁})... (1-x^{s_k})]^{-1}, expressions for which they derived. Wright [5] indicated a simpler method by which to find part of the solution (at least in the case $s_i=i$). The current paper gives a simple method by which the power-series expansion of a rational function may be derived. Lemma 1 is well known and gives the general form of the solution. Lemma 2 is also well known. See, for example, Andrews [1], Example 2, p. 98. Lemma 3 shows how the recurrence relation of Lemma 2 becomes of bounded degree in certain cases. The recurrence relation is then solved, and the solution is extended from these certain cases to all cases. We then apply the result to investigate the growth of the difference $P_S(n) - P_T(n)$, where S and T are finite sets, and in particular when this difference is bounded. The differences $P_S^{(0)}(n) - P_T^{(0)}(n)$ and $P_S^{(1)}(n) - P_T^{(1)}(n)$ are also considered, where $P_S^{(0)}$ (resp. $P_S^{(1)}$) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.
