New Euler–Mahonian Statistics on Permutations and Words

Abstract

We define new Mahonian statistics, called MAD, MAK, and ENV, on words. Of these, ENV is shown to equal the classical INV, that is, the number of inversions, while for permutations MAK has been already defined by Foata and Zeilberger. It is shown that the triple statistics (des, MAK, MAD) and (exc, DEN, ENV) are equidistributed over the rearrangement class of an arbitrary word. Here, exc is the number of excedances and DEN is Denert's statistic. In particular, this implies the equidistribution of (exc, INV) and (des, MAD). These bistatistics are not equidistributed with the classical Euler–Mahonian statistic (des, MAJ). The proof of the main result is by means of a bijection which, in the case of permutations, is essentially equivalent to several bijections in the literature (or inverses of these). These include bijections defined by Foata and Zeilberger, by Françon and Viennot and by Biane, between the symmetric group and sets of weighted Motzkin paths. These bijections are used to give a continued fraction expression for the generating function of (exc, INV) or (des, MAD) on the symmetric group.