Combinatorics on lattice paths in strips. (May 2021)

• Record Type: Journal Article
• Authors: Gu, Nancy S.S.; Prodinger, Helmut
• Is Part Of: European journal of combinatorics. Volume 94(2021)
• Publication Date: 2021-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103310
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: For lattice paths in strips which begin at ( 0, 0 ) and have only up steps U : ( i, j ) → ( i + 1, j + 1 ) and down steps D : ( i, j ) → ( i + 1, j − 1 ), let A n, k denote the set of paths of length n which start at ( 0, 0 ), end on heights 0 or − 1, and are contained in the strip − ⌊ k + 1 2 ⌋ ≤ y ≤ ⌊ k 2 ⌋ of width k, and let B n, k denote the set of paths of length n which start at ( 0, 0 ) and are contained in the strip 0 ≤ y ≤ k . We establish a bijection between A n, k and B n, k . The generating functions for the subsets of these two sets are discussed as well. Furthermore, we provide another bijection between A n, 3 and B n, 3 by translating the paths to two types of trees.