Babson-Steingrimsson statistics are indeed Mahonian (and sometimes even Euler-Mahonian)

Babson and Steingrimsson have recently introduced seven new permutation sta tistics, that they conjectured were all Mahonian (i.e., equi-distributed wi th the number of inversions). We prove their conjecture for the first four and also prove that the first and the fourth are even Euler-Mahonian. We us e two different, in fact, opposite, techniques. For three of them we give a computer-generated proof, using the Maple package ROTA, that implements th e second author's "Umbral Transfer Matrix Method." For the fourth one a geo metric permutation transformation is used that leads to a further refinemen t of this Euler-Mahonian distribution study. (C) 2001 Academic Press.