Walks confined in a quadrant are not always D-finite
Abstract
We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z^2, and always stay in the quadrant x >= 0, y >= 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- November 2002
- DOI:
- 10.48550/arXiv.math/0211432
- arXiv:
- arXiv:math/0211432
- Bibcode:
- 2002math.....11432B
- Keywords:
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- Mathematics - Combinatorics;
- 05A15 (primary)
- E-Print:
- To appear in Theoret. Comput. Sci. (special issue devoted to random generation of combinatorial objects and bijective combinatorics)