In their paper, Dokos et al. conjecture that the major index statistic is equidistributed among \(1423\)-avoiding, \(2413\)-avoiding, and \(3214\)-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, \(\Theta:S_n(1423)\to S_n(2413)\) and \(\Omega:S_n(2314)\to S_n(2413)\). In fact, we show that \(\Theta\) (respectively, \(\Omega\)) preserves numerous other statistics including the descent set, right-to-left maxima (respectively, left-to-right minima), and a statistic we call steps. Additionally, \(\Theta\) (respectively, \(\Omega\)) fixes all permutations avoiding both \(1423\) and \(2413\) (respectively, \(2314\) and \(2413\)).
Published: J. Combin. Theory Ser. A (124) 2014, 166-177, arXiv.