Simple formulas for lattice paths avoiding certain periodic staircase boundaries

TL;DR

This paper explores generalized formulas for counting lattice paths avoiding specific periodic staircase boundaries, extending classical results but noting limitations and open questions for broader generalizations.

Contribution

It identifies conditions under which simple formulas for lattice path counts extend to certain periodic staircase boundaries, highlighting limitations and open problems.

Findings

Classical formula for paths avoiding x=ky line is generalized to some periodic staircase boundaries.

The simple formula holds under specific conditions, but not universally.

Open questions remain about further generalizations of these formulas.

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.