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Class‐sum products in the symmetric group: Combinatorial interpretation of the reduced class coefficients
Author(s) -
Katriel Jacob
Publication year - 1998
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/(sici)1097-461x(1998)68:2<103::aid-qua3>3.0.co;2-z
Subject(s) - class (philosophy) , interpretation (philosophy) , product (mathematics) , mathematics , combinatorics , set (abstract data type) , group (periodic table) , multiplication (music) , representation (politics) , enumeration , direct product , combinatorial analysis , discrete mathematics , computer science , physics , quantum mechanics , artificial intelligence , politics , political science , law , programming language , geometry
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 sum [( p )] n that consists of a cycle of length p and n − p fixed points, with an arbitrary class sum in S n , was found to be expressible in terms of a set of reduced class coefficients (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 p , which has to do 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 evaluation of products involving [( p )] n for several values of n > p . The combinatorial 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. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 103–118, 1998