Polynomials, meanders, and paths in the lattice of noncrossing partitions
HTML articles powered by AMS MathViewer
- by David Savitt PDF
- Trans. Amer. Math. Soc. 361 (2009), 3083-3107 Request permission
Abstract:
For every polynomial $f$ of degree $n$ with no double roots, there is an associated family $\mathcal {C}(f)$ of harmonic algebraic curves, fibred over the circle, with at most $n-1$ singular fibres. We study the combinatorial topology of $\mathcal {C}(f)$ in the generic case when there are exactly $n-1$ singular fibres. In this case, the topology of $\mathcal {C}(f)$ is determined by the data of an $n$-tuple of noncrossing matchings on the set $\{0,1,\ldots ,2n-1\}$ with certain extra properties. We prove that there are $2(2n)^{n-2}$ such $n$-tuples, and that all of them arise from the topology of $\mathcal {C}(f)$ for some polynomial $f$.References
- Drew Armstrong. Generalized noncrossing partitions and combinatorics of Coxeter groups. http://arxiv.org/math.CO/0611106v1.
- Paul H. Edelman and Rodica Simion, Chains in the lattice of noncrossing partitions, Discrete Math. 126 (1994), no. 1-3, 107–119. MR 1264480, DOI 10.1016/0012-365X(94)90257-7
- Paul H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980), no. 2, 171–180. MR 583216, DOI 10.1016/0012-365X(80)90033-3
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- H. Tracy Hall. Meanders in a Cayley graph. http://arxiv.org/math.CO/0606170.
- G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350 (French). MR 309747, DOI 10.1016/0012-365X(72)90041-6
- S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoret. Comput. Sci. 117 (1993), no. 1-2, 227–241 (English, with English and French summaries). Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). MR 1235181, DOI 10.1016/0304-3975(93)90316-L
- Jeremy L. Martin, David Savitt, and Ted Singer, Harmonic algebraic curves and noncrossing partitions, Discrete Comput. Geom. 37 (2007), no. 2, 267–286. MR 2295058, DOI 10.1007/s00454-006-1283-6
- Jon McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006), no. 7, 598–610. MR 2252931, DOI 10.2307/27642003
- Rodica Simion, Noncrossing partitions, Discrete Math. 217 (2000), no. 1-3, 367–409 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766277, DOI 10.1016/S0012-365X(99)00273-3
Additional Information
- David Savitt
- Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
- Email: savitt@math.arizona.edu
- Received by editor(s): June 8, 2006
- Received by editor(s) in revised form: May 1, 2007, and June 18, 2007
- Published electronically: December 30, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3083-3107
- MSC (2000): Primary 05A18; Secondary 14P25
- DOI: https://doi.org/10.1090/S0002-9947-08-04579-0
- MathSciNet review: 2485419