On the maximal weight of $(p,q)$-ary chain partitions with bounded parts

Abstract: A $(p,q)$-ary chain is a special type of chain partition of integers with parts of the form $p^aq^b$ for some fixed integers $p$ and $q$. In this note, we are interested in the maximal weight of such partitions when their parts are distinct and cannot exceed a given bound $m$. Characterizing the cases where the greedy choice fails, we prove that this maximal weight is, as a function of $m$, asymptotically independent of $\max(p,q)$, and we provide an efficient algorithm to compute it.