A bijective proof of Jackson's 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. Our work 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. Key words: Symmetric group, factorizations, unicellular bicolored maps, permutations, bicolored trees, Harer-Zagier formula, Eulerian tours.

• Type: Journal
• Year: 2008
• Where: JCT
• Authors: Gilles Schaeffer, Ekaterina A. Vassilieva
• Keywords: JCT 2008, Ordered Bicolored Tree, Permutations, Symmetric Group