Congruences modulo powers of 2 for generalized Frobenius partitions with six colors

A generalized Frobenius partition of [Formula: see text] with [Formula: see text] colors is a two-rowed array [Formula: see text] where [Formula: see text], and the integer entries are taken from [Formula: see text] distinct copies of the non-negative integers distinguished by color, and the rows are ordered first by size and then by color with no two consecutive like entries in any row. Let [Formula: see text] denote the number of this kind of partitions of [Formula: see text] with [Formula: see text] colors. In this paper, we establish some congruences modulo powers of 2 for [Formula: see text].

CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.