Computational study of non-unitary partitions

Abstract

Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $\nu(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures regarding the growth of $\nu(n)$ as $n\to \infty$. Here we refine and prove some of these conjectures. For example, we prove $p(n) \sim \nu(n)\sqrt{n/\zeta(2)}$ as $n\to \infty$, and give Ramanujan-like congruences between $p(n)$ and $\nu(n)$ such as $p(5n)\equiv \nu(5n)\ (\operatorname{mod} 5)$.