A refinement of Wilf-equivalence for patterns of length 4

In their paper [6], 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, Θ:Sn(1423)→Sn(2413) and Ω:Sn(3214)→Sn(2413). In fact, we show that Θ (respectively, Ω) preserves numerous other statistics including the descent set, right-to-left maxima (respectively, left-to-right minima), and a statistic we call steps. Additionally, Θ (respectively, Ω) fixes all permutations avoiding both 1423 and 2413 (respectively, 3214 and 2413).