Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1148377 | Journal of Statistical Planning and Inference | 2014 | 14 Pages |
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 strictly 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 zn is replaced by a power series of the form znÏn(z), where Ïn(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. We give several concrete examples, including an alternative way to solve the tennis ball problem.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Joseph P.S. Kung, Anna de Mier, Xinyu Sun, Catherine Yan,