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Journal of the Ramanujan Mathematical Society

Volume 38, Issue 2, June 2023  pp. 121–128.

Computational study of non-unitary partitions

Authors:  A. P. Akande, Tyler Genao, Summer Haag, Maurice D. Hendon, Neelima Pulagam, Robert Schneider and Andrew V. Sills
Author institution: Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A.

Summary:  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) √ n/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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