Transitive factorisations into transpositions and holomorphic mappings on the sphere

We determine the number of ordered factorisations of an arbitrary permutation on $n$ symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on $n$ symbols. Such factorisations of the identity permutation have been considered by Crescimanno and Taylor in connection with a class of topologically distinct holomorphic maps on the sphere. As with Macdonald's construction for symmetric functions that multiply as the classes of the class algebra, essential use is made of Lagrange inversion.