小字母表超模式的渐近紧下界
An asymptotically tight lower bound for superpatterns with small alphabets
摘要：
如果置换$σ\in S_n$包含每个$τ\in S_k$作为一个模式，则它是$k$ -超模式（或$k$ -通用）.这种“超模式”的概念可以推广到小字母表上的单词，最近在Engen和Vatter的调查中提出了几个关于小字母表上超模式的问题。其中一个问题涉及$[k+1]$上最短的$k$ -超模式的长度。Miller的一个构造给出了$(k^2+k)/2$的一个上界，我们证明了它对低阶项是最优的。这意味着一个较弱版本的猜想埃里克森，埃里克森，Linusson和Wastlund。我们的结果也反驳了40年前Gupta的猜想。
英文摘要：
A permutation $σ\in S_n$ is a $k$-superpattern (or $k$-universal) if it contains each $τ\in S_k$ as a pattern. This notion of "superpatterns" can be generalized to words on smaller alphabets, and several questions about superpatterns on small alphabets have recently been raised in the survey of Engen and Vatter. One of these questions concerned the length of the shortest $k$-superpattern on $[k+1]$. A construction by Miller gave an upper bound of $(k^2+k)/2$, which we show is optimal up to lower-order terms. This implies a weaker version of a conjecture by Eriksson, Eriksson, Linusson and Wastlund. Our results also refute a 40-year-old conjecture of Gupta.
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