Sieve equivalence in generalized partition theory

Abstract

The sieve method (principle of inclusion-exclusion) is one of the most powerful tools of combinatorics. Given a set a of objects and a list 9 of properties of those objects, we determine the number N(Y; 2s) of objects in Q whose set of properties contains (those in 9 that are indexed by) the set S. We are then able to find the number of objects that have no properties, exactly k properties, etc. Now consider two lists a, 59 of properties of objects. We will say that the lists are sieve-equivalent, written a 9, if for every set S we have N(@; as> = N(9; ZS). (1) Clearly, if GZ 9, then all questions that the sieve method can answer will have the same answer relative to list GZ and to list 9. A form of this idea appears first in [8]. Daniel Cohen [ 1,2] has deduced many identities for the classical partition function in that way. The following argument is perhaps typical: If Q is the set of partitions of IZ, let Q!=(l+1,2+2,3+3,4+4 )... }, 9= { 2 , 4 , 6 , 8 ,... ].