A Formula for the Partition Function that Counts

Let p(n) denote the number of partitions of the integer n. The first exact formula for p(n) was published by Hardy and Ramanujan in 1918. Two decades later, Hans Rademacher improved the Hardy–Ramanujan formula to give an infinite series that converges rapidly to p(n). 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 it 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 formula for the partition function as a multisum each term of which actually counts a certain class of partitions. A comparison with a quasipolynomial representation will be given, as will as an associated polynomial approximation which appears to attain a level of accuracy comparable to that of the initial term of the Hardy–Ramanujan–Rademacher series