Tutte seminar - Bruce Richmond

Asymptotics of Some Partition Functions

Abstract:

Some recent results on the asymptotic analysis of partitions of integers will be discussed. First however some history of such problems will be briefly surveyed. The number, $p(m, n)$, of partitions of the integer $n$ into exactly $m$ parts will be studied. Euler showed that $$\sum_{n \ge 1} \sum_{m \ge 1} p(m, n)a^{m}z^{i} = \prod_{i \ge 1}\frac{1}{1 - az^{i}}.$$ There are many identities known between partition functions and much asymptotic analysis regarding these functions. A couple of examples will be described. Recently the number, $P(m,n)$ of plane partitions or two-dimensional partitions of $n$ with diagonal elements summing to $n$ has been studied. In this case $$\sum_{n \ge 1}\sum_{n \ge 1} P(m, n)a^{m}z^{n} = \prod_{i \ge 1} \frac{1}{\left(1 - az^{i}\right)^{i}}.$$ Using the saddle-point method with D. Panario and B. Young the asymptotic behaviour of $P(m, n)$ has been worked out. $P(m, n)$ satisfies a normal distribution. The details are unpleasantly intricate so only the barest sketch of the method which is widely applicable will be provided.