In a joint project, we are currently working on an online combinatorial statistic finder in which (beside other things) want to gather information about combinatorial collections and statistic, see the website <http://www.findstat.org>.


In this context, I am reading through Dominique Foata and Doron Zeilberger's paper


''Denert's permutation statistic is indeed Euler-Mahonian''


from 1990. In there, they prove that the bistatistic given by the number of exceedences and the Denert index is Euler-Mahonian. Moreover, they say:


"The most natural proof of this result would be in terms of a bijection from $\mathcal{S}\_n$ to itself that sends the pair (des,maj) simultaneously to the pair (exc,den). Although it is rather easy to find a bijetion that sends maj to den ..., and it is now trivial ... to find a bijection that sends exc to des, we are unable, at present, to find a bijection that does both at the same time. ... We really hope that such a bijective proof of Denert's conjecture will be found ... ."


So my question is:



> 
> Is there an explicit bijection on permutations known that sends thnumber of descents to the number of exceedences, and at the same time the major index to the Denert index?
> 
> 
> 



A bijective proof that the bistatistic $(\operatorname{exc},\operatorname{den})$ on permutations is Euler-Mahonian
Yes. 
G.-N. Han, Distribution Euler-mahonienne : une correspondance, C. R. Acad. Sci. Paris, 310, Série I, 1990, pp. 311-314.


G.-N. Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, 310, Série I, 1990, pp. 493-496.


G.-N. Han, Une transformation fondamentale sur les réarrangements de mots, Adv. in Math., 105(1), 1994, pp. 26-41.


