Random walk in a Weyl chamber
HTML articles powered by AMS MathViewer
- by Ira M. Gessel and Doron Zeilberger PDF
- Proc. Amer. Math. Soc. 115 (1992), 27-31 Request permission
Abstract:
The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge ${x_1} \geq {x_2} \geq \cdots \geq {x_n}$ (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps.References
-
D. André, Solution directe du problème résolu par M. Bertrand, C. R. Acad. Sci. Paris 105 (1887), 436-437.
- C. T. Benson and L. C. Grove, Finite reflection groups, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0383218
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588–621. MR 1503182, DOI 10.2307/1968753
- Michael Filaseta, A new method for solving a class of ballot problems, J. Combin. Theory Ser. A 39 (1985), no. 1, 102–111. MR 787720, DOI 10.1016/0097-3165(85)90085-8
- Michael E. Fisher, Walks, walls, wetting, and melting, J. Statist. Phys. 34 (1984), no. 5-6, 667–729. MR 751710, DOI 10.1007/BF01009436
- Frank G. Garvan, Some Macdonald-Mehta integrals by brute force, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 77–98. MR 1019845, DOI 10.1007/978-1-4684-0637-5_{8}
- Howard D. Grossman, Fun with lattice points, Scripta Math. 16 (1950), 207–212. MR 40257
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- David A. Huse and Michael E. Fisher, Commensurate melting, domain walls, and dislocations, Phys. Rev. B (3) 29 (1984), no. 1, 239–270. MR 729979, DOI 10.1103/physrevb.29.239
- Christian Krattenthaler, Enumeration of lattice paths and generating functions for skew plane partitions, Manuscripta Math. 63 (1989), no. 2, 129–155. MR 980569, DOI 10.1007/BF01168868
- C. Krattenthaler and S. G. Mohanty, $q$-generalization of a ballot problem, Discrete Math. 126 (1994), no. 1-3, 195–208. MR 1264487, DOI 10.1016/0012-365X(94)90264-X
- I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931
- I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988–1007. MR 674768, DOI 10.1137/0513070
- E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152, DOI 10.1007/BF01388841
- Robert A. Proctor, Reflection and algorithm proofs of some more Lie group dual pair identities, J. Combin. Theory Ser. A 62 (1993), no. 1, 107–127. MR 1198383, DOI 10.1016/0097-3165(93)90074-I
- T. Watanabe and S. G. Mohanty, On an inclusion-exclusion formula based on the reflection principle, Discrete Math. 64 (1987), no. 2-3, 281–288. MR 887367, DOI 10.1016/0012-365X(87)90197-X
- Doron Zeilberger, André’s reflection proof generalized to the many-candidate ballot problem, Discrete Math. 44 (1983), no. 3, 325–326. MR 696297, DOI 10.1016/0012-365X(83)90200-5
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 27-31
- MSC: Primary 05A15
- DOI: https://doi.org/10.1090/S0002-9939-1992-1092920-8
- MathSciNet review: 1092920