On the Parity of the Generalized Frobenius Partition Functions

Abstract

In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, φk(n) and cφk(n), which enumerate two types of combina- torial objects which Andrews called generalized Frobenius partitions. These objects are generalizations of Frobenius partitions, which are bijective to ordinary partitions. Ordinary partitions and their functions have been popularized by Ramanujan in the 20th century, which encouraged mathematicians to study them in the following years and to find what are called Ramanujan–like congruences. As part of his Memoir, Andrews proved a number of Ramanujan–like congruences satisfied by specific gener- alized Frobenius partition functions, and like Ramanujan, he paved the way for other authors to prove similar results for these functions, usually for a fixed parameter k. In this thesis, we gently introduce the reader to the ordinary partition and partition function, as well as the two families of generalized Frobenius partition functions that Andrews introduced. Our goal is to identify an infinite family of values of k such that φk(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, 1 ≤ r ≤ p − 1, such that 24r + 1 is a quadratic nonresidue modulo p,φpl−1(pn + r) ≡ 0 (mod 2) for all n ≥ 0. Such a result, which holds for φk(n) 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. We have also found and proven other results that pertain to generalized Frobenius partitions, which we will also be sharing in this thesis.