Some enumerative applications of cyclotomic polynomials

We begin with three formulas involving integer partitions, polynomials over F q , and Dirichlet series that have strong similarities.These formulas can be unified and extended by a general result involving factorization in a free monoid.The key fact underlying this approach is that for certain subsets S (called cyclotomic sets) of the positive integers, the numerator and denominator of a certain rational function G S (x) are products of cyclotomic polynomials.We then investigate properties of cyclotomic sets and conclude with a connection between certain cyclotomic sets and commutative algebra.