Abstract:

Recently, Ballantine and Welch considered two classes of integer partitions which they labeled POND and PEND partitions. These are integer partitions wherein the odd parts (respectively, the even parts) *cannot* be distinct. In recent work, I studied these two types of partitions from an arithmetic perspective and proved infinite families of mod 3 congruences satisfied by the two corresponding enumerating functions. I will talk about the generating functions for these enumerating functions, and I will also highlight the elementary proofs that I utilized.
In the latter portion of the talk, I will discuss unexpected connections between these divisibility properties for POND and PEND partitions by considering modified versions of the generating functions in question and relating these new generating functions in a natural way via Atkin-Lehner involutions. This part of the talk is joint work with Nicolas Smoot (University of Vienna).