Edelman and Greene constructed a bijective correspondence between the reduced words of the reverse permutation in the symmetric group Sn and standard Young tableaux of the staircase shape (n - 1, n - 2,..., 1). Our motivation originates from random sorting networks, a line of research initiated by Angel, Holroyd, Romik and Virág. We reformulate one of their conjectures on the shapes of intermediate configurations coming from random sorting networks. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.