We study the enumeration of different classes of grand knight's paths in the plane. In particular, we focus on the subsets of zigzag knight's paths subject to constraints. These constraints include ending at ordinate 0, bounded by a horizontal line, confined within a tube, among other considerations. We present our results using generating functions or direct closed-form expressions. We derive asymptotic results, finding approximations for quantities such as the probability that a zigzag knight's path stays in some area of the plane, or for the average of the final height of such a path. Additionally, we exhibit some bijections between grand zigzag knight's paths and some pairs of compositions.

翻译：我们研究了平面上各类大骑士路径的计数问题，特别关注受约束的锯齿骑士路径子集。这些约束包括终点纵坐标为0、受水平直线限制、局限在管道区域内等多种情况。我们通过生成函数或直接闭式表达式来呈现研究结果。推导出渐近结果后，得到了诸如锯齿骑士路径停留在平面某区域的概率、该类路径最终高度的平均值等量的近似值。此外，我们还展示了锯齿骑士路径与某些分拆对之间的双射关系。