A bijective proof of Jackson's formula for the number of factorizations of a cycle (Q941309)

In this paper factorizations of the cyclic permutation \((1 \,2\,\dots\,N)\) into two permutations with respectively \(n\) white and \(m\) black vertices are studied. These factorizations can be represented in a graphical way as so called unicellular partitioned bicolored maps. In Section 2 several enumeration formulas for these objects are listed, among which one is Jackson's and another is Adrianov's formula. These formulas were proved using heavy artillery like group characters and matrix integrals, but no elementary combinatorial proof was known. The authors of the present paper aim at such an elementary combinatorial proof and show that bicolored unicellular maps are in 1-1 correspondance with couples formed of an ordered tree and a partial permutation. Section 2 of the paper is devoted to show that left hand side and right hand side of Jackson's formula are connected to the two classes of objects mentioned earlier. Section 3 describes a bijection between the two mentioned classes of objests, and in Section 4 the map defined in Section 3 is proved to be a bijection indeed. Section 5 shows how Adrianov's formula can be obtained from Jackson's by tedious calculations using Vandermonde's conversion formulas. Section 6 finally is devoted to representing the described bijection in terms of Eulerian tours on directed graphs. In reading the paper, to really understand what is going on in Sections 3 and 4, one should work through an example following closely the described steps. It then turns out that the description is quite accurate but not easy to interpret. Doing the same for the material presented in Section 6 turned out to be impossible for me. The worked out examples in this Section are not correct. For instance in Figure 9 the permutation beta-bar should end in the cycles \((10\;12)(11)\) in stead of \((10\;11)(12)\) according to the formula given in the last but one paragraph of page 920. In Figure 10 the \((4\;5)\) cycle in psi should be replace by \((2\;5)\), since the edges corresponding to 4 and 5 are not in correspondence while those of 2 and 5 are. This would lead to another XI using the formula given at the bottom of page 923. Also I was not able to reconstruct Figure 11 based on the rather vague description given in the proof of Proposition 6.3. The material presented in Sections 3 and 4 was worthwhile studying and constitutes a nice elementary combinatorial proof of Jackson's formula.