从Hertzsprung问题到模式重写系统
From Hertzsprung's problem to pattern-rewriting systems
摘要：
利用Hertzsprung在1887年提出的一个问题，我们说一个给定的排列$π\in\mathcal{S}_n$包含Hertzsprung模式$σ\in\mathcal{S}_k$，如果有$π$的因子$π(d+1)π(d+2)\cdotsπ(d+k)$，使得$π(d+1)-σ(1) =\cdots = π(d+k)-σ(k)$。结合使用Goulden-Jackson聚类方法和转移矩阵方法，我们确定了任何一组（不可比的）Hertzsprung模式出现的联合分布，从而基本上推广了Jackson等人关于排列中上升和下降游程分布的早期结果。我们将我们的结果应用于计算排列到模式替换等价的问题，并使用模式重写系统-一种类似于研究较多的字符串重写系统的新形式-我们解决了Linton等人提出的几个开放问题。
英文摘要：
Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation $π\in\mathcal{S}_n$ contains the Hertzsprung pattern $σ\in\mathcal{S}_k$ if there is factor $π(d+1)π(d+2)\cdotsπ(d+k)$ of $π$ such that $π(d+1)-σ(1) =\cdots = π(d+k)-σ(k)$. Using a combination of the Goulden-Jackson cluster method and the transfer-matrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations. We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems -- a new formalism similar to the much studied string-rewriting systems -- we solve a couple of open problems raised by Linton et al. in 2012.
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