Abstract
Consider a single walker on the slit plane, that is, the square grid Z2 without its negative x-axis, who starts at the origin and takes his steps from a given set 6. Mireille Bousquet-Mélou conjectured that – excluding pathological cases – the generating function counting the number of possible walks is algebraic if and only if the walker cannot cross the negative x-axis without touching it. In this paper we prove a special case of her conjecture.
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References
Cyril Banderier and Philippe Flajolet, Basic analytic combinatorics of directed lattice paths, Theoret. Comput. Sci. 281 (2002), no. 1–2, 37–80, Selected papers in honour of Maurice Nivat. MR 2003g:05006
Mireille Bousquet-Mélou, Walks on the slit plane: other approaches, Advances in Applied Mathematics 27 (2001), no. 2–3, 243–288, Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000). MR 2002j:60076
Michael Drmota, A bivariate asymptotic expansion of coefficients of powers of generating functions, European Journal of Combinatorics 15 (1994), no. 2, 139–152. MR 94k:05014
Philippe Flajolet, Analytic models and ambiguity of context-free languages, Theoretical Computer Science 49 (1987), no. 2–3, 283–309, Twelfth international colloquium on automata, languages and programming (Nafplion, 1985). MR 89e:68067
Philippe Flajolet and Robert Sedgewick, The average case analysis of algorithms, 1994.
Einar Hille, Analytic function theory. Vol. I, II. 2nd ed. corrected., Chelsea Publishing Company, 1973.
Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 37 #1562
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Rubey, M. (2004). Transcendence of Generating Functions of Walks on the Slit Plane. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds) Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7915-6_6
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DOI: https://doi.org/10.1007/978-3-0348-7915-6_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9620-7
Online ISBN: 978-3-0348-7915-6
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