Counting Cores and Bar Cores: From Modular Forms to McKay numbers

Abstract

In this thesis, we delve into the theory of cores of partitions and tackle two main problems: the enumeration of t-cores and t-bar cores, and the computation of McKay numbers for the symmetric and alternating groups. For the enumeration problem, we discuss new and known explicit results for small values of t and bounds for general values of t which are obtained through the theory of modular forms. We also present new generating functions for t-bar cores in the case when t is even. For the computation of McKay numbers, we invoke the theory of p-core towers, for primes p, which serves as a direct application of the topic of enumeration of p-cores to other combinatorial problems. We also resolve the values for p=2 further and study them modulo 4. This thesis presents itself as a survey of existing results in literature that have paved the way towards solving the two problems and as a presentation of original contributions in the same direction.