NEW EULER-MAHONIAN STATISTICS ON PERMUTATIONS AND WORDS

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 numbe r of inversions, while for permutations MAK has been already defined b y Foata and Zeilberger. It is shown that the triple statistics (des, M AK, MAD) and (exc, DEN, ENV) are equidistributed over the rearrangemen t class of an arbitrary word. Here, exc is the number of excedances an d DEN is Denert's statistic. In particular, this implies the equidistr ibution of (exc, INV) and (des, MAD). These bistatistics are not equid istributed with the classical Euler-Mahonian statistic (des, MAJ). The proof of the main result is by means of a bijection which, in the cas e of permutations, is essentially equivalent to several bijections in the literature (or inverses of these). These include bijections define d by Foata and Zeilberger, by Francon and Viennot and by Biane, betwee n the symmetric group and sets of weighted Motzkin paths. These biject ions are used to give a continued fraction expression for the generati ng function of (exc, INV) or (des, MAD) on the symmetric group. (C) 19 97 Academic Press.