Minimal transitive products of transpositions: the reconstruction of a proof of A. Hurwitz

We want to draw the combintorialists attention to an important, but apparently little known paper by the function theorist A. Hurwitz, published in 1891, where he announces the solution of a counting problem which has gained some attention recently: in how many ways can a given permutation be written as the product of transpositions such that the transpositions generate the full symmetric group, and such that the number of factors is as small as possible (under this side condition). The function theoretic origin and interest of this problem will not be discussed in the present note — see the original paper by Hurwitz [14]. Current work on related problems is contained e.g. in the article by El Marraki et al. [9] in this volume