Georgia Institute of Technology College of Sciences

The number of permutations with specified descent set is maximized on the (classical) alternating permutations, which are counted by the Euler numbers. Asymptotics for the Euler numbers are well-studied. A counterpart of the descent statistic is the exceedance set statistic which is known to be maximized on a single run of consecutive exceedances. An exact enumeration is known, but the asymptotics have not been studied. We provide asymptotics using multivariate analytic combinatorics (providing a uniformity that goes beyond the range of a basic central limit theorem). This answers a question of E. Clark and R. Ehrenborg. As further applications we also discuss generalized pattern avoidance, and the number of orbit types (n-cycles) that admit a stretching pair (a certificate for "turbulence" in the context of combinatorial dynamics). This includes joint work with R. Ferraz de Andrade and B. Nagle and J. N. Cooper.