A reciprocity relation for t-designs (Q1112033)

It is shown that for a given t, v, k, the monoid of all t-(v,k,\(\lambda)\) designs is a toroidal monoid in the sense of Hochester and Stanley and the corresponding monoid algebra is Gorenstein. Among the consequences deduced is the fact that the function \(P^ v_{k,t}(\lambda)=\) the number of distinct t-(v,k,\(\lambda)\) designs is quasi-periodic and satisfies a reciprocity relation. Some generalizations and related problems are discussed.