CLASS-SUM PRODUCTS IN THE SYMMETRICAL GROUP - COMBINATORIAL INTERPRETATION OF THE REDUCED CLASS COEFFICIENTS

An algorithm for the evaluation of the structure constants in the clas s algebra of the symmetric group has recently been considered. The pro duct of the class sum [(p)](n) that consists of a cycle of length p an d II - p fixed points, with an arbitrary class sum in S-n, was found t o be expressible in terms of a set of reduced class coefficients (RCCs ), the p-RCCs. The combinatorial significance of the p-RCCs is elucida ted, showing that they are related to a well-defined enumeration probl em within S-p, which has to do with a certain refinement of the corres ponding class multiplication problem. This is in contrast with the rep resentation-theoretic evaluation of the p-RCCs, which requires the eva luation of products involving [(p)](n) for several values of n > p. Th e combinatorial interpretation of the p-RCCs allows the derivation of some of their previously conjectured properties and of some of the ''e limination rules'' that specific types of p-RCCs were found to satisfy . (C) 1998 John Wiley & Sons, Inc.