A new proof of the refined alternating sign matrix theorem

Abstract

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n × n alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0–1–(−1) matrices generalizing alternating sign matrices.