Ever since Viggo Brun's pioneering work, number theorists have develop
ed increasingly sophisticated refinements of the sieve of Eratosthenes
to attack problems such as the twin prime conjecture and Goldbachs co
njecture. Ever since Gian-Carlo Rota's pioneering work, combinatoriali
sts have found more and more areas of combinatorics where sieve method
s (or Mobius inversion) are applicable. Unfortunately, these two devel
opments have proceeded largely independently of each other even though
they are closely related. This paper begins the process of bridging t
he gap between them by showing that much of the theory behind the numb
er-theoretic refinements carries over readily to many combinatorial se
ttings. The hope is that this will result in new approaches to;nd more
powerful tools for sieve problems in combinatorics such as the comput
ation of chromatic polynomials, the enumeration of permutations with r
estricted position, rind the enumeration of regions in hyperplane arra
ngements. (C) 1998 Academic Press.