Some combinatorial applications of cyclotomic polynomials

A theorem of MacMahon states that the number of partitions of n for which no part appears exactly once equals the number of partitions of n into parts ≡ 1 (mod 6). The key fact behind this identity is that the numerator and denominator of a certain rational function are products of cyclotomic polynomials. We generalize MacMahon's result using the concept of a cyclotomic subset of the positive integers. We discuss properties of cyclotomic sets and show a connection to numerical semigroups (submonoids M of the nonnegative integers N such that N-M is finite) and commutative algebra. We then show how cyclotomic sets can be applied to other situations, in particular, counting polynomials over finite fields and evaluating Dirichlet series.