A New Bijective Proof of Babson and Steingrímsson's Conjecture

Abstract

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted stat. Given a permutation π, let des(π) denote the descent number of π and maj(π) denote the major index of π. Babson and Steingrímsson conjectured that (des, stat) and (des, maj) are equidistributed on S n . Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

“…In the same paper [2], Babson and Steingrímsson conjectured the bistatistic pdes, STATq is equidistributed with pdes, MAJq. This was first proved by Foata and Zeilberger [11], see also [4,8,12,20] for further developements along this line. Since clearly despp rc q " desppq for any permutation p, we include here the equivalent version for BAST, which will also be needed in Section 7.…”

“…Among the new Mahonian statistics introduced by Babson and Steingrímsson [2], the following one STAT " 21`132`213`321 (6.1) has attracted much attention (see for example [4,8,11,12,20]). The equidistribution results we have seen so far in this paper are between BAST " STAT rc and MAJ (see Corollaries 1.5 and 1.6), over S n p132q.…”

Equidistributions of MAJ and STAT over pattern avoiding permutations

“…In 2017, the author and Li [5] gave a new bijective proof of Babson and Steingrímsson's Conjecture by analogy with Carlitz's insertion method [4]. While this bijection does not preserve the statistic adj.…”

“…By Theorem 2.6 and Lemma 2.8, we give a bijective proof of Theorem 2.1. As mentioned in the introduction, there are two involutions on S n which map the statistic (maj, stat) to (stat, maj) given by Burstein [3] and Chen and Li [5], respectively. We note that α is not the restriction of either of them.…”

Equidistributions of Mahonian Statistics over Pattern Avoiding Permutations

“…These have since been proved and reproved at various levels of refinement by a number of authors (see e.g. [16,4,31,6]).…”