On directed lattice paths with vertical steps

详细信息    查看全文

• 作者：Maciej Dziemiańczuk ^{mdziemia@inf.ug.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Simple directed lattice paths ; Bijective combinatorics
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 March 2016
• 年：2016
• 卷：339
• 期：3
• 页码：1116-1139
• 全文大小：661 K

文摘

This paper is devoted to the study of non-simple directed lattice paths running between two fixed points and for which the set of allowed steps contains vertical step V=(0,−1)$V=(0,-1)$ and forward steps S_k=(1,k)$S_k=(1,k)$ for some k∈Z$k\in \mathbb{Z}$. These paths generalize the heavily-studied simple directed lattice paths that consist of only forward steps. Two special families of primary (restricted to the half-plane) and free (unrestricted) lattice paths are considered. It is shown that for any family of primary paths with vertical steps there is equinumerous family of proper weighted simple directed lattice paths. The relationship between primary and free paths is established and some combinatorial and statistical properties are obtained. Finally, four families of paths with vertical steps are presented and related to Łukasiewicz, Raney, Dyck, Motzkin, Schröder, and Delannoy paths.