AI 中文总结
本文引入k级联停车函数，推导了基于排列模式的递归公式，其结果序列可关联Motzkin数、Catalan数等，为相关卷积三角形行和提供新组合解释。
AI 中文摘要
我们引入了k级联停车函数，这是停车函数的参数化变体，其中汽车形成最多包含k≥0辆汽车的碰撞级联。当k=0时，得到经典停车函数；当k=1时，得到MVP停车函数。尽管停车函数和级联停车函数作为集合是等价的，但作为映射通常是不同的。因此，本文研究了它们结果纤维的计数问题。我们的主要结果是一个基于排列模式的递归公式，用于计算任意给定排列的纤维大小，适用于任意k≥0。当该公式应用于最长排列时，它会生成一个整数序列族，在最简单序列（k=0）、Motzkin数（k=1）和Catalan数（k≥n-1）之间插值。当应用于分层排列集合时，该公式为某些卷积三角形的行和提供了新的组合解释，包括Motzkin卷积三角形和Catalan卷积三角形。
英文摘要
We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.
Comments13 Pages, 2 Figures
