We define or redefine new Mahonian permutation statistics, called "mad," "mak" and "env." Of these, env is shown to equal the classical "inv," that is the number of inversions, while "mak" has been defined in a slightly different way by Foata and Zeilberger. It is shown that the triple statistics (des,mak,mad) and (exc,den,env) are equidistributed over the symmetric group. Here "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 is essentially equivalent to several bijections in the literature (or inverses of these). These include bijections defined by Foata and Zeilberger, by Francon 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.
@article{SLC_1995_35_a2,
author = {Robert J. Clarke and Einar Steingr{\'\i}msson and Jiang Zeng},
title = {New {Euler-Mahonian} {Permutation} {Statistics}},
journal = {S\'eminaire lotharingien de combinatoire},
year = {1995},
volume = {35},
url = {http://geodesic.mathdoc.fr/item/SLC_1995_35_a2/}
}
Robert J. Clarke; Einar Steingrímsson; Jiang Zeng. New Euler-Mahonian Permutation Statistics. Séminaire lotharingien de combinatoire, Tome 35 (1995). http://geodesic.mathdoc.fr/item/SLC_1995_35_a2/