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

TL;DR

This paper investigates the maximum total weight of distinct $(p,q)$-ary chain partitions with parts bounded by a limit, revealing asymptotic independence from $p$ and $q$, and offers an efficient computation method.

Contribution

It characterizes when the greedy approach fails and proves the asymptotic independence of the maximal weight from $ ext{max}(p,q)$, along with providing an efficient algorithm.

Findings

Maximal weight is asymptotically independent of $ ext{max}(p,q)$.

Characterization of cases where greedy algorithm fails.

Development of an efficient algorithm to compute the maximal weight.

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.