A bijective proof of Jacksonʹs formula for the number of factorizations of a cycle

Factorizations of the cyclic permutation ( 1 2 … N ) into two permutations with respectively n and m cycles, or, equivalently, unicellular bicolored maps with N edges and n white and m black vertices, have been enumerated independantly by Jackson and Adrianov using evaluations of characters of the symmetric group. In this paper we present a bijection between unicellular partitioned bicolored maps and couples made of an ordered bicolored tree and a partial permutation, that allows for a combinatorial derivation of these results. rk is closely related to a recent construction of Goulden and Nica for the celebrated Harer–Zagier formula, and indeed we provide a unified presentation of both bijections in terms of Eulerian tours in graphs.