| Summary: |
本論文研製之一個 Dyck path 是一條在第一象限內的格線路徑，以原點 (0,0) 為起點，使用向上步 (1,1) 與向下步 (1,−1)，終點落在 x 軸上。Dyck paths 的數目以 Catalan 來計數，Dyck paths 可以推廣到 m-Fuss paths。在m-Fuss paths 中，可使用的單位步向上步 (m,m) 和向下步 (1,−1)。在2010 年，Deutsch 等學者將 Dyck paths 的概念推廣到 skew Dyck paths。Skew Dyck path 是一個在第一象限的格線路徑，以原點 (0,0) 為起點，可使用的單位步有向上步 (1,1) ，右下步 (1,−1) 和左下步 (−1,−1) ，終點落在 x 軸上。Deutsch 等學者並發現了 skew Dyck paths 的許多性質。在本篇論文中，我們定義了 skew m-Fuss paths，可使用的單位步有向上步 (m,m) ，右下步 (1,−1) 和左下步 (−1,−1)。Skew m-Fuss paths 的定義可視為 m-Fuss paths 和 skew Dyck paths 的共同推廣。我們得到skew m-Fuss paths 的生成函數與計數公式。 It is well known that the number of Dyck paths, which are the paths in he first quadrant using up-steps (1,1), down-steps (1,−1), starting from (0,0) and ending on x-axis, is counted by the Catalan numbers. Dyck paths can be eralized to the m-Fuss paths, which the allowable steps are up-steps (m,m) and down-step (1,−1). It is also known that the number of m-Fuss paths with n up-steps is counted by the m-Fuss number. In 2010, the concept of Dyck paths is generalized to the skew Dyck paths by Deutsch et al. A skew Dyck path is a lattice path in the first quadrant using up-steps (1,1), down-steps (1,−1) and left-step (−1,−1), starting from (0,0) and ending on x-axis. Deutsch et al. enumerated the number of skew Dyck paths and found many properties. In this paper we define the skew Fuss paths, in which the allowable steps are up-steps (m,m), down-steps (1,−1) and left-step (−1,−1). Our definition of skew Fuss paths is a simultaneous generalization of the Fuss paths and skew Dyck paths. We enumerate the skew Fuss paths and derive some of their properties. |