### Volume 21 Issue 3 Paper #P3.37 — Comment by the Authors

**Sep 12, 2014, revised Sep 24, 2014**

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.