MacMahon’s Partial Fractions

Cayley used ordinary partial fractions decompositions of 1/[(1-x)(1-x^2). . .(1-x^m)] to obtain direct formulas for the number of partitions of n into at most m parts for several small values of m. No pattern for general m can be discerned from these, and in particular the rational coefficients that appear in the partial fraction decomposition become quite cumbersome for even moderate sized m. MacMahon gave a decomposition of 1/[(1-x)(1-x^2). . .(1-x^m)] into what he called "partial fractions of a new and special kind" in which the coefficients are "easily calculable number[s]" and the sum is indexed by the partitions of m. While MacMahon's derived his "new and special" partial fractions using "combinatory analysis," the aim of this talk is to give a preliminary report on a fully combinatorial explanation of MacMahon's decomposition. It seems likely that this will give a combinatorial explanation for the coefficients that appear in the ordinary partial fraction decompositions, which in turn can be used to give a formula for the number of partitions of n into at most m parts for arbitrary m.