Partitions and Sylvester waves

Abstract

The restricted partition function $$p_N(n)$$ counts the partitions of the integer n into at most N parts. In the nineteenth century, Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to $$p_N(n)$$ in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.