  We give manifestly positive Andrews-Gordon type series for the level 3
standard modules of the affine Lie algebra of type $A^{(1)}_2$. We also give
corresponding bipartition identities which have representation theoretic
interpretations via the vertex operators. Our proof is based on the Bordin
product formula and the Corteel-Welsh recursion for the cylindric partitions, a
$q$-version of Sister Celine's technique and a generalization of Andrews'
partition ideals by finite automata due to Takigiku and the author.
