Characterizing the Number of m-ary Partitions Modulo m

In this talk, we will focus our attention on $m$-ary partitions which are integer partitions wherein each part must be a power of a fixed integer $m>1$. Since the late 1960s, numerous mathematicians (including Churchhouse, Andrews, Gupta, Rodseth, and Sellers) have studied divisibility properties of m-ary partitions. After discussing some of these historical results, I will describe a novel and unexpected conjecture communicated to me by Aviezri Fraenkel which characterizes the divisibility of the ternary partition function $b_3(n)$ based on the base 3 representation of $n$. I will provide a proof of this result, and will close with a wonderful generalization that follows quite naturally. This is joint work with George Andrews and Aviezri Fraenkel.

James A. Sellers