A permutation $\sigma \in S_n$ is said to be k-universal or a k-superpattern if for every $\pi \in S_k$, there is a subsequence of σ that is order-isomorphic to π. A simple counting argument shows that σ can be a k-superpattern only if $n\ge (1/e^2+o(1))k^2$, and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this by designing an efficient encoding scheme for the patterns that appear in σ. This approach is quite flexible and is applicable to other universality-type problems; for example, we also improve a bound by Engen and Vatter on a problem concerning $(k+1)$-ary sequences which contain all k-permutations.

排列 $\sigma \in {\mathrm{小号}}_{ñ}$如果每一个都被称为k通用或k超级模式$\pi \in {\mathrm{小号}}_{ķ}$，有一个σ的子序列，它与π是同构的。一个简单的计数参数表明，σ可以是ķ -superpattern仅当$ñ\ge （\mathrm{1个}/{Ë}^{\mathrm{2个}}+Ø（\mathrm{1个}））{ķ}^{\mathrm{2个}}$，并且Arratia推测此下限是最佳可能。证明Arratia的猜想，我们通过一个小的常数因子来改善琐碎的界限。我们通过为σ中出现的模式设计有效的编码方案来实现这一点。这种方法非常灵活，适用于其他普遍性问题。例如，我们还改善了Engen和Vatter关于以下问题的界限$（ķ+\mathrm{1个}）$包含所有k个排列的-ary序列。