Combinatorics on lattice paths in strips
NSS Gu, H Prodinger�- European Journal of Combinatorics, 2021 - Elsevier
NSS Gu, H Prodinger
European Journal of Combinatorics, 2021•ElsevierFor 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�…
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�…
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.
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