Partitions into a small number of part sizes

Document Type

Article

Publication Date

2-1-2017

Abstract

© 2017 World Scientific Publishing Company. We study νk(n), the number of partitions of n into k part sizes, and find numerous arithmetic progressions where ν2 and ν3 take on values divisible by 2 and 4. Expanding an earlier work, we show ν2(An + B) ≡ 0 (mod 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 p(n) ≡ 0 (mod 16) in the first three progressions (the fourth is known), and thereby show that ν3(An + B) ≡ 0 (mod 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.

Publication Title

International Journal of Number Theory

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