Abstract

Asinowski, Bacher, Banderier and Gittenberger [1] recently developed the vectorial kernel method – a powerful extension of the classical kernel method useable for paths that obey constraints that can be described by finite automata, e.g. avoid a fixed pattern, avoid several patterns at once, stay in a horizontal strip and many others more. However, they only considered walks with steps of length one. In this paper we will generalize their results to walks with longer steps. We will also give some applications of this extension and prove a conjecture about the asymptotic behavior of the expected number of ascents in Schröder paths.

摘要

Asinowski、Bacher、Banderier 和 Gittenberger [1] 最近开发了矢量核方法 - 经典核方法的强大扩展，可用于遵循可由有限自动机描述的约束的路径，例如避免固定模式，一次避免多个模式，留在水平地带等等。然而，他们只考虑步长为 1 的步行。在本文中，我们将把他们的结果推广到步数更长的步行。我们还将给出这个扩展的一些应用，并证明关于 Schröder 路径中预期上升次数的渐近行为的猜想。