Partitions into a small number of part sizes

Abstract

Abstract. We study ν k (n), the number of partitions of n into k part sizes, and find numerous arithmetic progressions where ν 2 and ν 3 take on values divisible by 2 and 4. Expanding earlier work, we show ν 2 (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 ν 3 (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.

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

“…. ., and an s m -core partition (see [1,19]). For instance, Figure 1 gives the Young diagram and hook lengths of the partition (6, 3, 2, 1) and Figure 2 gives the Young diagram and hook lengths of its conjugation.…”

“…Figure 5 corresponds to the partition (22,17,12,12,9,9,9,6,6,6,3,3,3,3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1). Figure 6 corresponds to the partition (24,19,14,10,10,10,7,7,7,4,4,4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1). Both of them have the size 135 and are conjugate to each other.…”

“…for example, S = {1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 21, 22, 27, 33} is not the β-set of a(6,17,19)-core partition, since 22 − 17 = 5 ̸ ∈ S, which violates Condition (2) in Theorem 16. Observe that (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) = (4, 4, 4, 2, 1, 1) and (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ) = (4, 4, 4, 4, 3, 3).…”