AN ALGEBRAICALLY DERIVED Q-ANALOG OF A CHARACTER SUM ASSOCIATED WITH A CLASS OF SEMIREGULAR PERMUTATIONS

The group algebra of the symmetric group can be used to determine the cycle structure of permutations which are obtained as products of designated conjugacy classes. Such matters arise, for example, in certain topological questions and in the embedding of graphs on orientable surfaces. We consider a set of permutations restricted by cycle structure, and use basic hypergeometric series to derive q-analogues associated with the generating functions for the numbers of such permutations. The expressions which are derived pose a number of combinatorial questions about their connexion with the Hecke algebra of the symmetric group. © 1990 by Pacific Journal of Mathematics.