标题： 关于加权和有界多维Catalan数 显示英文标题

标题： On Weighted and Bounded Multidimensional Catalan Numbers

摘要： 我们定义了多维Catalan数的加权类似物，得到了其中一些的基于矩阵的递推关系，并给出了它们周期性的条件。 在此框架基础上，我们引入了两个新的三角形数组序列：第一个序列枚举了精确高度为$s$的$k$维平衡投票路径；第二个序列是Narayana数的一个新的多维推广，它计算具有恰好$p$个峰的平衡投票路径的数量。 显示英文摘要

摘要： We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of triangular arrays: the first one enumerates the $k$-dimensional Balanced ballot paths of exact height $s$; the second one is a new multidimensional generalization of the Narayana numbers, which count the number of Balanced ballot paths with exactly $p$ peaks.