Lattice and Schröder paths with periodic boundaries |
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Authors: | Joseph PS Kung Anna de Mier Xinyu Sun Catherine Yan |
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Institution: | 1. Department of Mathematics, University of North Texas, Denton, TX 76203, USA;2. Centre de Recerca Matemàtica, 08193 Bellaterra, Spain;3. Department of Applied Mathematics 2, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034 Barcelona, Spain;4. Mathematics Department, Tulane University, 6823 St. Charles Avenue, New Orleans, LA 70118, USA;5. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;6. Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China |
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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 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 znφn(z), where φn(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. |
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Keywords: | primary 05A15 secondary 05A10 05A40 |
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