Class-sum products in the symmetric group: Combinatorial interpretation of the reduced class coefficients

✍ Scribed by Jacob Katriel

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 214 KB
• Volume: 68
• Category: Article
• ISSN: 0020-7608
• DOI: 10.1002/(sici)1097-461x(1998)68:2<103::aid-qua3>3.0.co;2-z

✦ Synopsis

An algorithm for the evaluation of the structure constants in the class algebra of the symmetric group has recently been considered. The product of the class wŽ .x sum p that consists of a cycle of length p and n y p fixed points, with an arbitrary n class sum in S , was found to be expressible in terms of a set of reduced class coefficients n Ž . RCCs , the p-RCCs. The combinatorial significance of the p-RCCs is elucidated, showing that they are related to a well-defined enumeration problem within S , which has to do p with a certain refinement of the corresponding class multiplication problem. This is in contrast with the representation-theoretic evaluation of the p-RCCs, which requires the wŽ .x evaluation of products involving p for several values of n ) p. The combinatorial n interpretation of the p-RCCs allows the derivation of some of their previously conjectured properties and of some of the ''elimination rules'' that specific types of p-RCCs were found to satisfy.