Computational study of non-unitary partitions

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Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let ν(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 ν(n) as n → ∞. Here we refine and prove some of these conjectures. For example, we prove p(n) ∼ ν(n) pn/ζ(2) as n → ∞, and give Ramanujan-like congruences between p(n) and ν(n) such as p(5n) ≡ ν(5n) (mod 5).

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Journal of the Ramanujan Mathematical Society

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