By National Council of Teachers of Mathematics
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Extra resources for Enrichment Mathematics for the Grades 27th Yearbook
Gessel and D. Zeilberger, Random walk in a Weyl chamber, Proc. Amer. Math. Soc. 115 (1992) 27-3l.  I. P. Goulden and D. M. Jackson, Combinatorial enumeration, John Wiley and Sons, 1983.  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 .  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.  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.  A. M. Odlyzko. Asymptotic enumeration methods, volume II. Elsevier, 1995. In Handbook of Combinatorics, (R. Graham, M. Grotschel, and L. ).  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 . 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).
Enrichment Mathematics for the Grades 27th Yearbook by National Council of Teachers of Mathematics