Revisiting Rademacher's Formula for the Partition Function p(n

Research in Number Theory, 2019

Special functions 2000: current perspective and future …, 2001

2013

This expository article describes recent work by the authors on the partition function p(n). This includes a finite formula for p(n) as a "trace" of algebraic singular moduli, and an overarching � -adic structure which controls partition congruences modulo powers of primes � ≥ 5.

Proceedings of The American Mathematical Society, 2007

The Ramanujan Journal, 2012

Let p r,s (n) denote the number of partitions of a positive integer n into parts containing no multiples of r or s, where r > 1 and s > 1 are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for p r,s (n). Keywords q-series • partitions • circle-method • Hardy-Ramanujan-Rademacher Mathematics Subject Classification (2000) Primary 11P82 • Secondary 05A17 • 11L05 • 11D85 • 11P55 • 11Y35 1 Introduction A partition of a positive integer n is a representation of n as a sum of positive integers, where the order of the summands does not matter. We use p(n) to denote the number of partitions of n, so that, for example, p(4) = 5, since 4 may be represented as 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. The function p(n) increases rapidly with n, and it is difficult to compute p(n) directly for large n. Rademacher [23], by slightly modifying earlier work of Hardy and Ramanujan [13], derived a remarkable infinite series for p(n). To describe this series

Research in Number Theory, 2021

The minimal excludant, or "mex" function, on a set S of positive integers is the least positive integer not in S. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely pt,t(n), and provide complete parity characterizations of p 1,1 (n) and p 3,3 (n). In this article, we study the parity of pt,t(n) when t = 2 α , 3 • 2 α for all α ≥ 1. We prove that p 2 α ,2 α (n) and p 3•2 α ,3•2 α (n) are almost always even for all α ≥ 1. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo 2 satisfied by p 2 α ,2 α (n) and p 3•2 α ,3•2 α (n) for all α ≥ 1.

Consideration of a classification of the number of partitions of a natural number according to the members of sub-partitions differing from unity leads to a non-recursive formula for the number of irreducible representations of the symmetric group S n .

International Mathematics Research Notices

arXiv: Combinatorics, 2018

In this note, we provide a simple derivation of expressions for the restricted partition function and its polynomial part. Our proof relies on elementary algebra on rational functions and a lemma that expresses the polynomial part as an average of the partition function.