On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$

Abstract

In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, φ k (n) and cφ k (n), which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter k. In this brief note, our goal is to identify an infinite family of values of k such that φ k (n) is even for all n in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers ℓ, all primes p ≥ 5, and all values r, 0 < r < p, such that 24r + 1 is a quadratic nonresidue modulo p, φ pℓ−1 (pn + r) ≡ 0 (mod 2) for all n ≥ 0. Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical q-series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of k, is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.