The number of lattice paths below a cyclically shifting boundary

TL;DR

This paper extends classical lattice path enumeration by counting paths under cyclically shifting boundaries, providing new formulas and refinements for paths with specified corners, with applications to periodic boundary-dominated paths.

Contribution

It introduces a novel enumeration formula for lattice paths under cyclically shifting boundaries, extending classical results and including a refinement for counting paths with a certain number of corners.

Findings

Derived a new enumeration formula for cyclically shifting boundaries

Provided a refinement for counting paths with a fixed number of corners

Applied results to paths dominated by periodic boundaries

Abstract

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical reflection argument. A refinement allows for the counting of paths with a specified number of corners. We also apply the result to examine paths dominated by periodic boundaries.