The Multi-Generating Function for Intervals in Young's Lattice: some Comments on a Paper by Azam and Richmond

Azam and Richmond studied the generating function \(P_\lambda(y)\), which enumerates (by length) partitions in the lower ideal \([0,\lambda]\) in the Young lattice. They found a rational recursion for \[Q_k(\mathbf{x},y) = \sum_{\lambda \in \Lambda(k)} P_\lambda(y) \mathbf{x}^{\lambda}.\] We show that their results can be extended to a multi-graded version. By interpreting the original problem as one of enumerating plane partitions with two rows, we can describe the multi-graded version of \(Q_k\) using the integer transform of a certain rational pointed polyhedral cone. We furthermore relate Azam's and Richmond's result to those obtained by Andrews and Paule using MacMahon's \(\Omega\)-operator.