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Babson–Steingrı́msson Statistics are Indeed Mahonian (and Sometimes Even Euler–Mahonian)

2001; Elsevier BV; Volume: 27; Issue: 2-3 Linguagem: Inglês

10.1006/aama.2001.0741

1090-2074

Dominique Foata, Doron Zeilberger,

semigroups and automata theory

Babson and Steingrı́msson have recently introduced seven new permutation statistics, that they conjectured were all Mahonian (i.e., equi-distributed with 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 use two different, in fact, opposite, techniques. For three of them we give a computer-generated proof, using the Maple package ROTA, that implements the second author's “Umbral Transfer Matrix Method.” For the fourth one a geometric permutation transformation is used that leads to a further refinement of this Euler–Mahonian distribution study.