AI 中文总结
本文证明一致随机置换矩阵的 Füredi-Hajnal 和 Stanley-Wilf 极限以高概率至多为 exp(O(√k(log k)^{5/2}))，改进先前上界，并结合下界表明对数阶为 k^{1/2+o(1)}。
AI 中文摘要
我们证明，阶为 $k$ 的一致随机置换矩阵的 Füredi-Hajnal 极限和 Stanley-Wilf 极限以趋于 1 的概率（当 $k\to\infty$ 时）至多为 $\exp\bigl(O(\sqrt{k}(\log k)^{5/2})\bigr)$。这改进了 Cibulka 和 Kynčl 的界 $\exp\bigl(O(k^{2/3}(\log k)^{7/3}/(\log\log k)^{1/3})\bigr)$。结合 Fox 给出的下界，这些界表明对于几乎所有置换，这两个极限的对数均为 $k^{1/2+o(1)}$。
英文摘要
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.
