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Gessel and D. Zeilberger, Random walk in a Weyl chamber, Proc. Amer. Math. Soc. 115 (1992) 27-3l. [23] I. P. Goulden and D. M. Jackson, Combinatorial enumeration, John Wiley and Sons, 1983. [24] D. J. Grabiner, Random walk in an alcove of an affine Weyl group, and noncolliding random walks on an interval, in Proceedings of FPSAC 2001, Tempe, Arizona, pp. 217- 225 . [25] R. Guy, B. E. Sagan and C. Krattenthaler, Lattice paths, reflections, and dimension-changing bijections, Ars Combin. 34 (1992) 3-15.

Discrete Mathematics, 218:167- 183, 2000. [20] Donatella Merlini, Renzo Sprugnoli, and Maria Cecilia Verri. An algebra for generating trees. In Mathematics and computer science (Versailles, 2000), pages 127- 139, Basel, 2000. Birkhiiuser. [21] A. M. Odlyzko. Asymptotic enumeration methods, volume II. Elsevier, 1995. In Handbook of Combinatorics, (R. Graham, M. Grotschel, and L. ). [22] Elisa Pergola, Renzo Pinzani, and Simone Rinaldi. Towards an algebra of succession rules. In Mathematics and computer science (Versailles, 2000), pages 141- 152, Basel, 2000.

This book contains one example for which the series P(x, y) is algebraic: no surprise, the steps of the corresponding walk are exactly Kreweras' steps ... This result is actually due to Flatto and Hahn [19]. The equation satisfied by the series P(x, y) does not work exactly like the equations for complete generating functions like Q(x, y; t): roughly speaking, the third variable t is replaced by the additional constraint P(x, y) = 1. Very recently, I have found a new, simpler proof of Flatto and Hahn's result (at least, in the symmetric case).

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Enrichment Mathematics for the Grades 27th Yearbook by National Council of Teachers of Mathematics


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