2020/04/01 by Gu, Nancy S. S., Prodinger, Helmut · 1 citation
#05A19 #05C05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.00684
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 An,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 -\lfloor(k+1)/(2)\rfloor ≤ y ≤ \lfloor(k)/(2)\rfloor of width k, and let Bn,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 An,k and Bn,k. The generating functions for the subsets of these two sets are discussed as well. Furthermore, we provide another bijection between An,3 and Bn,3 by translating the paths to two types of trees.