Congruences for overpartitions with restricted odd differences
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Congruences for overpartitions with restricted odd differences Michael D. Hirschhorn1 · James A. Sellers2 Received: 20 October 2018 / Accepted: 18 February 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t(n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t(n) satisfies an elegant congruence modulo 3, namely, for n ≥ 1, t(n) ≡
(−1)k+1 0
(mod 3) if n = k 2 for some integer k, (mod 3) otherwise.
In this work, using elementary tools for manipulating generating functions, we prove that t satisfies a corresponding parity result. We prove that, for all n ≥ 1, t(2n) ≡
1 (mod 2) if n = (3k + 1)2 for some integer k, 0 (mod 2) otherwise.
We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t(n) for various moduli. Keywords Congruences · Overpartitions · Restricted odd differences Mathematics Subject Classification 05A17 · 11P83
B
James A. Sellers [email protected] Michael D. Hirschhorn [email protected]
1
School of Mathematics and Statistics, UNSW, Sydney 2052, Australia
2
Department of Mathematics, Penn State University, University Park, PA 16802, USA
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M. D. Hirschhorn, J. A. Sellers
1 Introduction In a recent work, Bringmann et al. [2] defined the function t(n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. For example, t(4) = 8, where the overpartitions in question are given by the following: 4, 4, 3 + 1, 3 + 1, 2 + 2, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. By considering certain q-difference equations, the authors prove that the generating function for t(n) is given by f3 t(n)q n = , f1 f2 n≥0
where f k := (1 − q k )(1 − q 2k )(1 − q 3k ) . . . They also proved that t(n) satisfies an elegant congruence modulo 3. Theorem 1.1 For all n ≥ 1, (−1)k+1 t(n) ≡ 0
(mod 3) if n = k 2 for some integer k, (mod 3) otherwise.
In this work, using elementary tools for manipulating generating functions, we prove that t satisfies a corresponding parity result. Theorem 1.2 For all n ≥ 1, 1 (mod 2) if n = (3k + 1)2 for some integer k, t(2n) ≡ 0 (mod 2) otherwise. We also provide a truly elementary proof of the mod 3 characterization provided by Bringmann et al., as well as proofs of a number of additional congruences satisfied by t(n) for various moduli. We list these additional congruences here: Theorem 1.3 For all n ≥ 0, t(24n + 4) ≡ 0
(mod 4),
(1)
t(32n + 4) ≡ 0 t(48n + 36) ≡ 0
(mod 4), (mod 4).
(2) (3)
t(16n + 14) ≡ 0
(mod 12),
(4)
Theorem 1.4 For all n ≥ 0,
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Congruences for overpartitions with restricted odd differences
t(24n + 22
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