COMBINATORIAL DATA OF CERTAIN LATTICE PATHS

Abstract. In this paper, we will consider certain lattice paths in the two-dimensional space R2, satisfying the so-called axis property: if a lattice path starts at the point (0, 0) and ends on the horizontal axis (or the x-axis) of R2, then we will say that this lattice path satisfies the axis property. We find the cardinalities of the collection of all length-n lattice paths with the axis property. We show that those values are determined by the well-known Pascal’s Triangle, recursively. In the context, we provide the applications of this computations. In particular, those cardinalities explain the spectral property of certain groupoids having the fractal property (See [9] and [10]). In this paper, we will observe lattice paths embedded in the 2-dimensional space R2, understood as the coordinate system with the corresponding horizontal axis (or the x-axis) and the corresponding vertical axis (or the y-axis), as usual. This work is originally motivated by the recent research of the last author and Jorgensen’s joint work (See [9] and [10]). We will not consider the results of [9] and [10] in this paper, but in the following paragraph we briefly sketch the main stream o