Congruences for overpartitions with restricted odd differences

摘要

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) = similar to (-1)k+1 (mod 3) if n = k2 for some integer k, 0 (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) = similar to 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.