A Pentagonal Number Sieve

We prove a general pentagonal sieve theorem that has corollaries such as the following First the number of pairs of partitions of n that have no parts in common is p n p n p n p n p n Second if two unlabeled rooted forests of the same number of vertices are chosen i u a r then the probability that they have no common tree is Third if f g are two monic polynomials of the same degree over the eld GF q then the probability that f g are relatively prime is q We give explicit involutions for the pentagonal sieve theorem generalizing earlier mappings found by Bressoud and Zeilberger The Main Theorem The natural context in which our results lie is that of prefabs A prefab P is a combinatorial structure in which each object is uniquely representable as a product synthesis of powers of prime objects and in which there is an order function j j Z which satis es j j j j j j We denote the primes of P by p p Examples of prefabs are integer partitions rooted unlabeled forests plane partitions etc Let P be a prefab in which the number of objects of order n is f n for n and the number of prime objects of order n is bn for n The unique factorization of all objects in P into products of Supported in part by NSF grant DMS Supported in part by NSF grant DMS Supported in part by the O ce of Naval Research Supported in part by the National Science Foundation powers of prime objects is expressed by the formula X