Title: 'Euler's partition theorem; combining combinatorics and algebra'
Abstract: 'In 1748 the prolific Swiss mathematician L. Euler noted that the number of partitions of n into odd summands equals the number of partitions of n into distinct summands. Although Euler's theorem has a trivial algebraic proof, it also has many interesting combinatorial proofs. These involve constructing explicit bijections between the two sets of partitions. Euler's theorem allows one to enumerate certain 'representations' of symmetric groups and related groups. In order to study these representations, one needs partition bijections with special properties. Constructing new bijections with these properties sheds new light on both the representations and on Euler's theorem.
In summary, I hope to illustrate how algebraic combinatorics (the study of finite structures and counting using the methods of abstract algebra) enriches both algebra and combinatorics. No prior knowledge of partitions or groups is needed.'