SUMMER SEMINAR SERIES in Combinatorics and Experimental Math

Title: A Revival of Sylvester's Wave Theory of Partitions

Speaker: Drew Sills, Georgia Southern University and DIMACS

Date: Monday, July 11, 2011 5:00pm

Location: CoRE Bldg, CoRE 301, Rutgers University, Busch Campus, Piscataway, NJ


Let $p(n)$ denote the number of partitions of the integer $n$. The first exact formula for $p(n)$ was given by Hardy and Ramanujan in 1918. Their formula is a divergent series, which can be truncated in the appropriate place to give the exact value of $p(n)$. Rademacher improved Hardy and Ramanujan's formula in 1938 to a rapidly converging infinite series. In early 2011, Ono and Bruinier announced a new formula which expresses $p(n)$ as a finite sum of algebraic numbers.

The Hardy-Ramanujan-Rademacher formula is really a statement about the coefficients of a certain modular form whose coefficents happen to be the values of $p(n)$. The Ono-Bruinier formula expresses $p(n)$ as a sum of singular moduli of a certain weak Maass form that can be described in terms of the Dedekind eta function and the quasimodular Eisensten series $E_2$. Although $p(n)$ is clearly a combinatorial function, neither of these formulas is combinatorial.

In this talk, I will attempt to show that J. J. Sylvester (1857) and J. W. L. Glaisher (1909) were well on their way to finding a combinatorial formula for $p(n)$, and that a revival of their work combined with the power of modern computers could lead to a new formula for $p(n)$.