  Azam and Richmond arXiv:2107.09149 obtained a recursion for the generating
function of \(P_\lambda(y)\), itself a generating function enumerating by
length partitions in the lower ideal \([0,\lambda]\) in the Young lattice. We
show that this recursion can be extended to a multi-graded version. This is
done by interpreting the original problem as enumerating plane partitions with
two rows. We can then use the well-developed theory of lattice points in
polyhedral cones to determine some properties of the generating function. We
also relate Azam and Richmond's result to those obtained by Andrews and Paule
using MacMahons Omega-operator.
