Typical growth of the Füredi-Hajnal and Stanley-Wilf limits

Source abstract

We prove that the Füredi-Hajnal limit and the Stanley-Wilf limit of a uniformly random permutation matrix of order $k$ are at most $\exp\bigl(O(\sqrt{k}(\log k)^{5/2})\bigr)$ with probability tending to one as $k\to\infty$. This improves the bound $\exp\bigl(O(k^{2/3}(\log k)^{7/3}/(\log\log k)^{1/3})\bigr)$ of Cibulka and Kynčl. Together with the lower bound due to Fox, these bounds show that the logarithms of both limits are $k^{1/2+o(1)}$ for almost all permutations.