On the association scheme of perfect matchings and their designs

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph K 2 n . We show that they are special subsets of the association scheme obtained from the Gelfand pair ( S 2 n , S 2 ≀ S n ) . This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra ℂ [ S 2 n ] and using the representation theory of S 2 n .