Class-sum products in the symmetric group: Minimally-connected reduced class coefficients

The class algebra of the symmetric group plays an important role in the study of systems of identical particles. Recent progress in an ongoing effort to develop an algorithm for the combinatorial evaluation of the corresponding structure constants is presented. A connection is established with a closed form expression for a certain set of structure constants, obtained by Goupil and Bédard and by Goulden and Jackson. This allows the evaluation of a corresponding set of reduced coefficients that, in turn, have considerably wider applicability. Some outstanding open problems are presented.