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Integer Partitions

Updated: Mar 11, 2021

Say you want to know how many ways you can write n as a sum of positive integers, up to reordering the summands. The number of ways is the nth partition number p(n). 100 years ago, Ramanujan discovered three surprising divisibility properties, he discovered that p(5n+4) is always divisible by 5, p(7n+5) is always divisible by 7, and p(11n+6) is always divisible by 11. Trying to understand these and similar congruences led to important work on newforms by Atkin and Lehner and we now know that many similar congruences come about from the theory of Galois representations attached to modular forms.

A lot of cool math has been developed, just trying to understand the partition numbers, and understanding the partition numbers is what a lot of my research does. If we put these numbers into a generating function, the resulting function has lots of symmetries and is a modular form. Using the theory of modular forms, I have studied integer partitions both in terms of their asymptotic properties and arithmetic properties, including work towards understanding the Ramanujan-type congruences that can occur.

The partition function p(n) counts the nonincreasing sequences of positive integers whose sum is n. Despite having such an elementary definition, partition numbers are connected to deep mathematics. The Ramanujan congruences are three simple divisibility patterns for partition numbers, and understanding them led to some of the major developments in the theory of modular forms throughout the twentieth century. I am interested in a collection of statistical and arithmetic aspects of partitions.

One question I’m interested in is classifying the types of Ramanujan-type congruences that can occur, more specifically, finding a description for the triples (A,B,p) such that p(An+B)=0(mod p) for all n. For example, are there any of the form (Q*P, B, P), where Q, P are primes, and P is at least 13? This is not known, but my work with Ahlgren and Raum suggests that such triples, if they exist, are rare in a certain sense. Surprisingly, some undergraduate students that I supervised last fall found that such congruences do in fact exist when you look at colored partition numbers. This is a surprising finding, and in my future research I’m hopeful that understanding the difference between the colored and uncolored situations will help us answer the question for p(n).

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