The Non-existence and Scarcity of Congruences for Partitions

We investigate Ramanujan congruences for the function $\overline{t}(n)$, which counts the overpartitions of $n$ with restricted odd differences, and the existence of certain congruences of for $p_r(n)$. In particular, we show that one Ramanujan congruence exists for $\overline{t}(n)$ and that congruences of the form $p_r(\ell Q^m + \beta)$ for $\ell, Q$ prime and $m = 1,2$ appear to be scarce. The method for both results uses the theory of modular forms. In the former case, a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients is proved, which generalizes work done by Jonah Sinick. In the latter case, we develop several necessary conditions for the existence of such congruences, which generalizes the work of Ahlgren et. al. for $p(n)$.