A Formula for the Partition Function that "Counts"

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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.


Armstrong Atlantic State University Discrete Mathematics Seminar


Savannah, GA

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