On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern

Consider, for a permutation $\sigma \in {\cal S}_k$, the number $F(n,\sigma)$ of permutations in ${\cal S}_n$ which avoid $\sigma$ as a subpattern. The conjecture of Stanley and Wilf is that for every $\sigma$ there is a constant $c(\sigma) We also discuss $n$-permutations, containing all $\sigma \in {\cal S}_k$ as subpatterns. We prove that this can be achieved with $n=k^2$, we conjecture that asymptotically $n \sim (k/e)^2$ is the best achievable, and we present Noga Alon's conjecture that $n \sim (k/2)^2$ is the threshold for random permutations.