Enumeration of Alternating Sign Matrices

An alternating sign matrix (ASM) is a square matrix with entries in {0, 1, −1} whose nonzero entries alternate in sign along every row and every column, beginning and ending with +1. They emerged from Dodgson condensation, and the question of how many there are of each size stood open for more than a decade before Zeilberger and Kuperberg answered it in 1996. This graduate-level course develops the machinery behind the proof of Kuperberg and follows it into the current literature, where lattice models, symmetric function identities, and characters of the classical groups meet a family of objects — ASMs, plane partitions, monotone triangles — that are equinumerous with each other for reasons still not fully understood. The course has two major themes: symmetry classes via the six-vertex model, and refined enumeration together with the bijective questions that remain open.