Construction and Generation

Abstract

We have counted the orbits of finite groups on finite sets and successively refined our methods by introducing enumeration by weight as well as enumeration by stabilizer class. Moreover we discussed actions on structured sets like posets and semigroups. Later on the permutation group representations were refined by introducing linear representations, which led to applications in both directions. It remains to discuss the most difficult problem, the construction of a transversal of the orbits. We shall briefly discuss the general case of this problem, in order to introduce the concept of Sims chains, for cases when the acting group is given by generators and relations. We shall then use it in a detailed description of a direct evaluation of a transversal of the orbits of G on Y ^X with prescribed content λ. After that we describe a recur­sive method, using recursion on |Y|, and combining this recursion with the orderly generation method that was introduced by R. C. Read.