有理斜率线下格点路径的核方法
The kernel method for lattice paths below a line of rational slope
摘要：
本文分析了有理斜率线下格路的一些计数性质和渐近性质。我们说明了我们的方法与戴克路径下的一条线的斜率$2/5$。这回答了Knuth在2014年6月在巴黎举行的“算法分析”会议（AofA'2014）期间的“Flajolet讲座”中提出的问题#4。我们的方法扩展了Banderier和Flajolet的工作的渐近性和枚举的有向格路径的情况下，生成函数涉及几个占主导地位的奇点，并已应用到一类完整的问题，涉及一些“周期性”。证明中的一个关键要素是Knuth自己的一个老技巧的推广（用于枚举堆栈可排序的排列），由Flajolet和其他人推广为“内核方法”。所有相应的生成函数都是代数的，它们提供了一些新的组合恒等式，这些恒等式也可以用Wilf-Zeilberger-Petkovsek的A=B精神来处理。我们展示了如何获得类似的结果，任何合理的斜率。一个有趣的情况是，例如，戴克路径低于斜率$2/3$（这对应于所谓的杜雄俱乐部模型），我们解决了一个猜想有关的渐近的区域低于这样的晶格路径。我们的工作还提供了访问以下的非理性斜率（例如，戴克路径低于$y=x/\sqrt{2}$），一个问题，我们研究的同伴文章的晶格路径。
英文摘要：
We analyse some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope $2/5$. This answers Knuth's problem #4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities, and has applications to a full class of problems involving some "periodicities". A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovsek. We show how to obtain similar results for any rational slope. An interesting case is e.g. Dyck paths below the slope $2/3$ (this corresponds to the so-called Duchon's club model), for which we solve a conjecture related to the asymptotics of the area below such lattice paths. Our work also gives access to lattice paths below an irrational slope (e.g. Dyck paths below $y=x/\sqrt{2}$), a problem that we study in a companion article.
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