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An Eulerian partner for inversions. (English) Zbl 0982.05006
Summary: A number of researchers studying permutation statistics on the symmetric group $$S_n$$ have considered pairs $$(x, Y)$$, where $$x$$ is an Eulerian statistic and $$Y$$ is a Mahonian statistic. Of special interest are pairs such as (des, MAJ), whose joint distribution on $$S_n$$ is given by Carlitz’s $$q$$-Eulerian polynomials. We present a natural Eulerian statistic stc such that the pair (stc, INV) is equally distributed with (des, MAJ) on $$S_n$$, and provide a simple bijective proof of this fact. This result solves the problem of finding an Eulerian partner for the Mahonian statistic INV. We conjecture several properties of the joint distributions of stc with the statistics des and MAJ.

##### MSC:
 05A15 Exact enumeration problems, generating functions
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