Symmetric functions and Macdonald’s result for top connexion coefficients in the symmetric group. (English) Zbl 0830.20021

Goulden, I. P.; Jackson, D. M.

J. Algebra 166, No. 2, 364-378 (1994).

Summary: Macdonald (unpublished) gave an indirect proof that the connexion coefficients for certain symmetric functions coincide with the connexion coefficients of the class algebra of the symmetric group. We give a direct proof of this result and demonstrate the use of these functions in a number of combinatorial questions associated with ordered factorisations of permutations into factors of specified cycle-type, including factorisations considered up to commutation in the symmetric group. Several related properties of the symmetric functions are given.

MSC:

20C30	Representations of finite symmetric groups
05E05	Symmetric functions and generalizations
20B30	Symmetric groups

Keywords:

symmetric functions; connexion coefficients; class algebras; symmetric groups; ordered factorisations of permutations