Partitions into a small number of part sizes

TL;DR

This paper investigates the divisibility properties of partition functions into a small number of part sizes, revealing new arithmetic progressions where these functions are divisible by specific powers of 2, and explores related overpartition congruences.

Contribution

It extends previous work by identifying new arithmetic progressions with divisibility properties for $ u_2$ and $ u_3$, and proves novel congruences for overpartition functions.

Findings

$ u_2(An+B) ot o 0 mod 4$ for specific (A,B) pairs

Overpartition function $ar{p}(n) o 0 mod 16$ in certain progressions

$ u_3(An+B) o 0 mod 2$ in these progressions

Abstract

We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0 \pmod{4}$ for (A,B) = (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 $\bar{p}(n) \equiv 0 \pmod{16}$ in the first three progressions (the fourth is known), and thereby show that $\nu_3(An+B) \equiv 0 \pmod{2}$ 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.