# Combinatorial problems with stretched exponential asymptotics

## Seminar/Forum

Evan WIlliams Theatre
Peter Hall Building

We look at a range of combinatorial problems where the growth of coefficients is of the form $C.\mu^n\cdot \exp(-\alpha n^{\beta}) \cdot n^g,$ with $\alpha > 0,$ $0 < \beta < 1.$ Problems include some pattern-avoiding permutations, pushed" random walks,pushed" self-avoiding walks and interacting partially-directed walks. We will discuss, in a hand-waving way, how this stretched-exponential term arises, and give a numerical method for estimating the parameters $\mu,$ $\alpha,$ $\beta$ and $g.$ This new method arises from some recent investigations in number theory with Richard Brent and Larry Glasser. As an example, we give more precise asymptotics for the coefficients of $Av(1324)$ pattern-avoiding permutations.