After this article appeared, the authors learned that the correspondence they use to move from permutations to Motzkin paths is due to Foata and Zeilberger, from the article "Denert's permutation statistic is indeed Euler-Mahonian" in Stud. Appl. Math. 83 (1990), pp. 31-59. Moreover, a paper of Clarke, Steingrimsson, and Zeng studies a statistic, Edif, that coincides with depth for permutations. See "New Euler-Mahonian statistics on permutations and words" in Adv. Appl. Math. 18 (1997), pp. 237-270. Theorem 10 of that paper, with $x=p=1$, yields our Equation 5.1, while its Corollary 11 with $x=p=1$ yields our Equation 5.2.