In his 1984 AMS Memoir, George Andrews defined the family of k-colored generalized Frobenius partition functions. These functions count two-rowed arrays of positive integers of equal length which naturally generalize Frobenius symbols of integer partitions. These are denoted by \(c\phi _k(n)\) where \(k\ge 1\) is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all \(n\ge 0,\) \(c\phi _2(5n+3) \equiv 0{\pmod {5}}.\) Soon after, many authors proved congruence properties for various k-colored generalized Frobenius partition functions. In 2009, Drake considered a further generalization of the Frobenius symbol and defined a new family of colored generalized Frobenius partition functions which count variants of the two-rowed arrays that Andrews defined in 1984. The key difference between this new family of functions and those defined by Andrews is that the variant allows for the rows of each array to be of different length. In this work, we focus specifically on Drake’s 2-colored generalized Frobenius partition function, and we prove a number of congruences satisfied by this particular function. Our proofs are truly elementary, relying on the corresponding generating function, classical q-series results, and elementary generating function manipulations.
