Edelman and Greene constructed a bijective correspondence between the reduced words of the reverse permutation (n, n - 1, …, 2, 1) and standard Young tableaux of the staircase shape (n - 1, …, 1). Recently, motivated by random sorting networks, we studied this bijection and discovered some new properties in joint work with Svante Linusson. In this talk, I will discuss them and, if time permits, also a related project with Linusson and Robin Sulzgruber on random sorting networks where the intermediate permutations avoid the pattern 132.