On the refined 3-enumeration of alternating sign matrices

Abstract

An explicit expression for the numbers $A(n,r;3)$ describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, $A(n,r;3)$'s are represented as 1-fold sums which can also be written in terms of terminating ${}_4{F}_3$ series of argument 1/4.

Some comments and references added. To appear in the David Robbins memorial issue of Advances in Applied Mathematics