Relations Involving Lattice Paths and Certain Sequences of Integers
Douglas R. Stocks · The Fibonacci Quarterly · 1967
Relations involving certain special planar lattice paths and certain sequences of integers have been studied previously [ 1] , [ 2]. We will state c e r-tain basic definitions which pertain to these studies, develop additional r e s u l t s involving other planar lattice paths, and finally, indicate generalizations of these r e s u l t s for lattice paths in k dimensional space. For convenience of reference some of the definitions a r e collected together and presented in P a r t 1. The remaining material will be found in P a r t 2. P a r t 1 In Euclidean k-dimensional space the set X of points such that p belongs to X if and only if each coordinate of p is an integer is called the unit lattice of that space. The statement that P is a lattice path in a certain space means that P is a sequence such that 1) each t e r m of P is a member of the unit lattice of that space, and 2) if X is a t e r m of P and Y is the next t e r m of P and x ^ and yi a r e the i coordinates of X and Y respectively, then]x ^- y j = 1 or 0 and for some j, |XJ- y j | = 1. If each of X and Y is a point of the unit lattice in Euclidean k-dimensional space, then the statement that the lattice path P is a path from X to Y means that P is finite, X is the first t e r m of P, and Y is the last t e r m of P0 If P is a lattice path, X is a t e r m of P, and Y is the next t e r m of P, then by the step \\XtY] of P is meant the line interval whose end points a r e X and Y. A lattice path P in Euclidean 2 or 3-space is said to be s y m m e t r i c with respect to the line k if and only if it is true that if X is a point of some step of P, then either X is a point of k or there exists a point Y of some step of P such that k is the perpendicular bisector of the line interval [X, Y]. Suppose that S = [ (xi9yt), (x2,y2)] i s a step of some lattice path P in Euclidean 2-space9 S is said to be x-increasing if x2-xt = 1 and x-decreasing