# A bijective proof that the bistatistic $(\operatorname{exc},\operatorname{den})$ on permutations is Euler-Mahonian

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?

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