Computer Algebra and Applications Seminar, Summer Semester 2025

Title:  On the Unexpected Connections in the Arithmetic Properties of POND and PEND Partitions

Speaker: Prof. James Sellers, University of Minnesota Duluth, U.S.A.

Time and Location: 
Wed, April 9, 2025, 2 p.m.; 
RISC Seminarroom, Castle of Hagenberg

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).