SYMMETRICAL FUNCTIONS AND MACDONALD RESULT FOR TOP CONNECTION COEFFICIENTS IN THE SYMMETRICAL GROUP

In unpublished work, Macdonald gave an indirect proof that the connexi on coefficients for certain symmetric functions coincide with the conn exion coefficients of the class algebra of the symmetric group. We giv e a direct proof of this result and demonstrate the use of these funct ions in a number of combinatorial questions associated with ordered fa ctorisations of permutations into factors of specified cycle-type, inc luding factorisations considered up to commutation in the symmetric gr oup. Several related properties of the symmetric functions are given. (C) 1994 Academic Press, Inc.