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

An old theorem of Sylvester states that a product of k consecutive positive integers each exceeding k is divisible by a prime greater than k. We shall give a proof of this theorem and apply it to prove a result of Schur on the irreducibility of certain polynomials.