Linked partition ideals and the Alladi–Schur theorem

Let S denote the set of integer partitions into parts that differ by at least 3, with the added constraint that no two consecutive multiples of 3 occur as parts. We derive trivariate generating functions of Andrews–Gordon type for partitions in S with both the number of parts and the number of even parts counted. In particular, we provide an analytic counterpart of Andrews' recent refinement of the Alladi–Schur theorem.