Advances in the Theory of Cores and Simultaneous Core Partitions

Abstract

The theory of s-core partitions, integer partitions whose hook sets avoid hooks of length s, lies at the intersection of a surprising number of fields, including number theory, combinatorics, and representation theory. A more recent trend has been to study partitions whose hook sets avoid multiple lengths, known as simultaneous core partitions. This paper, divided into five sections, is a review of five recent papers in this area by undergraduates ([3], [4], [5], [18], [67]). All of the authors surveyed conducted their research while participating in the University of Minnesota Duluth REU.In the first section, we introduce partitions, the abacus, s-core partitions, and their connections to several fields. In the second section, we turn to self-conjugate s-core partitions and discuss several theorems of L. Alpoge on their asymptotic behavior and their connection, for small s, with points on curves. In the third section, we discuss simultaneous (s, t)-core partitions and the work of A. Aggarwal and V. Wang on the Armstrong conjecture. The fourth section highlights results of A. Aggarwal, A. Berger, and V. Wang on the enumeration, weight, and containment properties of simultaneous (s, t, u)-core partitions. In the final section, we mention some areas of ongoing research connected to the work discussed here.The techniques used across these papers, ranging from generating functions and modular forms to more combinatorial tools such as abaci, posets, and lattice paths, give a flavor of the richness of the subject. We provide illustrative examples when full proofs are too lengthy.