The multi-generating function for intervals in Young's lattice: some comments on a paper by Azam and Richmond

Jan Snellman¹

¹Department of Mathematics, Linköping University

The electronic journal of combinatorics, Tome 32 (2025) no. 3

Résumé

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.

Détail

DOI : 10.37236/13847

Classification : 05A15, 05A17
Mots-clés : enumeration of plane partitions with two rows, average partition of fixed length, generating functions