  Answering a question of Clark and Ehrenborg (2010), we determine asymptotics
for the number of permutations of size n that admit the most common excedance
set. In fact, we provide a more general bivariate asymptotic using the
multivariate asymptotic methods of R. Pemantle and M. C. Wilson. We also
consider two applications of our main result. First, we determine asymptotics
on the number of permutations of size n which simultaneously avoid the
generalized patterns 21-34 and 34-21. Second, we determine asymptotics on the
number of n-cycles that admit no stretching pairs.
