Dazhao Tang: On a question of Ray and Chakraborty, 93-108

Abstract:

Let $\overline{A}_\ell(n)$ denote the number of $\ell$-regular overpartitions of $n$. Quite recently, Ray and Chakraborty investigated the arithmetic density properties on powers of primes satisfied by $\overline{A}_\ell(n)$. Utilizing an algorithm of Radu and Sellers, they proved a congruence modulo $7$ for $\overline{A}_7(n)$. Moreover, they stated without proof a congruence modulo $5$ for $\overline{A}_5(n)$ and a congruence modulo $11$ for $\overline{A}_{11}(n)$. For these three congruences, they asked for an elementary proof. In this paper, we establish six congruence families for these three partition functions, three of which are the corresponding generalizations of three congruences considered by Ray and Chakraborty.

Key Words: Partitions, $\ell$-regular overpartitions, congruences, dissections, internal congruences.

2020 Mathematics Subject Classification: Primary 11P83; Secondary 05A17.

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