Partitions into a small number of part sizes

Abstract

We study [Formula: see text], the number of partitions of [Formula: see text] into [Formula: see text] part sizes, and find numerous arithmetic progressions where [Formula: see text] and [Formula: see text] take on values divisible by 2 and 4. Expanding an earlier work, we show [Formula: see text] for ([Formula: see text]) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function [Formula: see text] in the first three progressions (the fourth is known), and thereby show that [Formula: see text] in each of these progressions as well, and discuss the relationship between these congruences in more generality. We end with open questions in this area.