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arXiv · 0709.1717

Lattice and Schroder paths with periodic boundaries

Joseph P. S. Kung·Anna de Mier·Xinyu Sun·Catherine H. Yan
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Abstract

We consider paths in the plane with $(1,0),$ $(0,1),$ and $(a,b)$-steps that start at the origin, end at height $n,$ and stay to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most $b/a,$ then the ordinary generating function for the number of such paths ending at height $n$ is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or ``umbral'' generating function, in which the power $z^n$ is replaced by a power series of the form $z^n ϕ_n(z),$ where $ϕ_n(0) = 1.$ Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic.

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Joseph P. S. Kung, Anna de Mier, Xinyu Sun, Catherine H. Yan. 2007-09-27. Lattice and Schroder paths with periodic boundaries. https://arxiv.org/abs/0709.1717

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