An Eulerian partner for inversions
Abstract: A number of authors studying permutation statistics on the symmetric group have considered pairs (x, Y), where x is an Eulerian statistic and Y is a Mahonian statistic. Of special interest are pairs involving the statistics des, exc, MAJ, and INV, which arise often in combinatorics. One pair of statistics which has a particularly nice joint distribution on S_n is (des, MAJ). A second pair, (exc, DEN), was shown to be equidistributed with des, MAJ) by Foata and Zeilberger in 1989. This left open the problem of finding a natural Eulerian statistic z such that (z, INV) is equidistributed with (des, MAJ) and (exc, DEN). We will define such a statistic z, and give a simple bijective proof that the pairs of statistics are equidistributed.