Combinatorics on lattice paths in strips. (May 2021) - Gu, Nancy S.S.; Prodinger, Helmut
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.