Abstract
Just as Eulerian numbers generalize to Coxeter groups, so too do the Narayana numbers. There is a great deal of interest in these generalizations. Entire books could be (and have been) dedicated to the subject. We give a brief survey of this circle of ideas in this chapter, with an emphasis on parallels with our discussion of the classical case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Armstrong D. Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem Am Math Soc. 2009;202(949):x+159. Available from: http://dx.doi.org/10.1090/S0065-9266-09-00565-1.
Brady T, Watt C. K(π, 1)’s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). vol. 94; 2002. p. 225–50. Available from: http://dx.doi.org/10.1023/A:1020902610809.
Brändén P. On linear transformations preserving the Pólya frequency property. Trans Am Math Soc. 2006;358(8):3697–716 (electronic). Available from: http://dx.doi.org/10.1090/S0002-9947-06-03856-6.
Carter RW. Conjugacy classes in the Weyl group. Compositio Math. 1972;25:1–59.
Fomin S, Reading N. Root systems and generalized associahedra. In: Geometric combinatorics. vol. 13 of IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI; 2007. p. 63–131.
Fomin S, Zelevinsky A. Y -systems and generalized associahedra. Ann Math (2). 2003;158(3):977–1018. Available from: http://dx.doi.org/10.4007/annals.2003.158.977.
Hersh P. Deformation of chains via a local symmetric group action. Electron J Combin. 1999;6:Research paper 27, 18 pp. (electronic). Available from: http://www.combinatorics.org/Volume_6/Abstracts/v6i1r27.html.
Reading N. Clusters, Coxeter-sortable elements and noncrossing partitions. Trans Am Math Soc. 2007;359(12):5931–58. Available from: http://dx.doi.org/10.1090/S0002-9947-07-04319-X.
Reiner V. Non-crossing partitions for classical reflection groups. Discrete Math. 1997;177(1-3):195–222. Available from: http://dx.doi.org/10.1016/S0012-365X(96)00365-2.
Shephard GC, Todd JA. Finite unitary reflection groups. Can J Math. 1954;6:274–304.
Simion R, Ullman D. On the structure of the lattice of noncrossing partitions. Discrete Math. 1991;98(3):193–206. Available from: http://dx.doi.org/10.1016/0012-365X(91)90376-D.
Solomon L. Invariants of finite reflection groups. Nagoya Math J. 1963;22:57–64.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Petersen, T.K. (2015). W-Narayana numbers. In: Eulerian Numbers. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3091-3_12
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3091-3_12
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4939-3090-6
Online ISBN: 978-1-4939-3091-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)