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In this thesis we study the question of certain sequences of ratios products of factorials are always integers. Equivalently, this is a study of when certain step functions related to the Beurling--Nyman criterion for the Riemann Hypothesis are always nonnegative. We give a complete classification of what we call the "height 1" case, which proves a conjecture of V. I. Vasyunin regarding certain step functions that only take the values 0 and 1. For larger height we give partial results: we prove a conjecture of A. Borisov that, for fixed height, the range of values taken by one of the step functions we consider increases with its length; additionally, we prove that for larger heights there exists a classification similar to that for height 1. In addition to the application of these theorems to the classification of integer factorial ratios and nonnegative step functions related to the Beurling--Nyman criterion, through work of A. Borisov, these results have applications to the classification of cyclic quotient singularities. These results are proved using a variety of methods. These include Beukers and Heckman's classification of algebraic hypergeometric functions, complex analysis techniques familiar to analytic number theory, and a theorem of Jim Lawrence about closed subgroups of the torus.