Congruences for a class of eta-quotients and their applications

In Mestrige (Res Number Theory 6(1), Paper No. 5, 2020), we proved an infinite family of congruences for this partition function for l = 11. In this paper, we extend the ideas that we have used in Mestrige (2020) to prove infinite families of congruences for the partition function p([1cld])(n) modulo powers of l for any integers c and d, for primes 5 \u3c= l \u3c= 17. This generalizes Atkin, Gordon and Hughes\u27 congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions on the congruence subgroup Gamma(0)(l). Finally we use these congruences to prove congruences and incongruences for l-colored generalized Frobenius partitions, l-regular partitions, and l-core partitions for l = 5, 7, 11, and 13. We also prove incongruences for 17-regular, and 17-core partitions