The Combinatorics behind Number-Theoretic Sieves

Abstract

Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated refinements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or Möbius inversion) are applicable. Unfortunately, these two developments have proceeded largely independently of each other even though they are closely related. This paper begins the process of bridging the gap between them by showing that much of the theory behind the number-theoretic refinements carries over readily to many combinatorial settings. The hope is that this will result in new approaches to and more powerful tools for sieve problems in combinatorics such as the computation of chromatic polynomials, the enumeration of permutations with restricted position, and the enumeration of regions in hyperplane arrangements.