Simple formulas for lattice paths avoiding certain periodic staircase boundaries

Abstract

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer.We show that the natural generalization of this simple formula continues to hold when the line x = ky is replaced by certain periodic staircase boundaries-but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized.

The number of lattice paths below a cyclically shifting boundary

“…As another interesting example we present the following result, also recently discovered independently by other authors [3]. (It appears there in a very slightly modified form.…”

“…This result is also implicit in Tamm [9,Propositions 2,3], where it appears in generating series form. The proof given there follows a probabilistic argument (originally due to Gessel) reliant on Lagrange inversion, whereas our derivation is purely combinatorial.…”

“…Notice that in each of the top three rows of the figure there are a total of 36 dominated paths from (0, 0) to (6,3). There are this many also in the bottom row if the three identical cyclic shifts of a = (2, 2, 2) are taken into account.…”

Lattice paths: Coding and complexity

“…Theorem [13]: A general result was formulated in [6]. The authors define as A w,v the infinite staircase path that starts at (0, v), then takes w steps east, v steps north, w steps east, v steps north, and so on.…”

“…Further, we shall demonstarte that this generating function method also allows to derive the identities from [6], which were obtained by a bijective argument. Finally, some open problems concerning further generalizations of the above numbers and th complexity to obtain these numbers or their reduction modulo 2 will be discussed.…”