## Transitive factorizations in the symmetric group, and combinatorial
aspects of singularity theory

**Ian Goulden**

Department of Combinatorics and Optimization

University of Waterloo

Abstract: Consider ordered factorizations of an arbitrary element of
the symmetric group on n symbols, into factors which all belong to the
same conjugacy class. An expression for the number of such factorizations
is available via characters, but in certain cases a more compact answer
can be obtained using a family of symmetric functions constructed by Macdonald.
With the further restriction that the group generated by the factors acts
transitively on the n symbols, we call these transitive factorizations.
Hurwitz considered the number of transitive factorizations in the case
that the factors are transpositions, because of its connection with counting
all nonequivalent ramified coverings of a Riemann surface. In this case
he gave a remarkably compact, explicit answer. In this talk, extensions
of Hurwitz' result are presented, suggesting that transitive factorisations
have an elegant structure, including a close, but as yet unknown, link
with Macdonald's symmetric functions.