A Formula for the Partition Function that Counts

An asymptotic formula for p(n), precise enough to give the exact value, was given by Hardy and Ramanujan in 1918. Twenty years later, Hans Rademacher improved the Hardy-Ramanujan formula to give an infinite series that converges to p(n). The Hardy-Ramanujan-Rademacher series is revered as one of the truly great accomplishments in the field of analytic number theory. In 2011, Ken Ono and Jan Bruinier surprised the world by announcing a new formula which attains p(n) by summing a finite number of complex numbers which arise in connection with the multiset of algebraic numbers that are the union of Galois orbits for the discriminant −24n + 1 ring class field. Thus despite the fact that p(n) is a combinatorial function, the known formulas for p(n) involve deep mathematics, and are by no means “combinatorial” in the sense that they involve summing a finite or infinite number of complex numbers to obtain the correct positive integer value. In this talk, I will present a combinatorial multisum expression for D(n, k), the number of partitions of n with Durfee square of order k. Of course, summing D(n, k) over 1 ≤ k ≤ √ n yields p(n). This, in turn leads to a natural approximation to p(n) as a polynomial with rational coefficients. Numerical evidence suggests that this polynomial approximation obtains accuracy comparable to that of the initial term of the Hardy-Ramanujan-Rademacher series. The idea behind the formula is due to Yuriy Choliy, and the work was completed in collaboration with him