In this talk I will investigate various combinatorial models which arise in the study of Coxeter groups. (This is joint work with Brant C. Jones.)

A stepping-off point will be the notion of one-line notation; when we write a finite permutation, we often write it in one-line notation as 15243. I will discuss a generalization for affine permutations, and then bijections with other combinatorial families such as abacus models, core partitions, and bounded partitions. If time permits, I will discuss an application of the abacus model—the enumeration of fully commutative affine permutations.

No previous understanding of Coxeter groups is necessary to enjoy this talk. A healthy appetite for interesting combinatorics will suffice.