Noncrossing partitions in surprising locations

American Mathematical Monthly

Volumn 113, Issue 7, 2006, Pages 598-610

  McCammond, Jon ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748308747     PISSN: 00029890     EISSN: None     Source Type: Journal    
DOI: 10.2307/27642003     Document Type: Article

References (51)

1. M. Aigner, Catalan and other numbers: A recurrent theme, in Algebraic Combinatorics and Computer Science, Springer Italia, Milan, 2001, pp. 347-390.
2. M. Anshelevich, Free stochastic measures via noncrossing partitions, Adv. Math. 155 (2000) 154-179.
3. _, Free stochastic measures via noncrossing partitions II, Pacific J. Math. 207 (2002) 13-30.
4. C. A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005) 179-196.
5. _, On noncrossing and nonnesting partitions for classical reflection groups, Electron. J. Combin. 5 (1998), 16 pp. (electronic).
6. C. A. Athanasiadis and S. Linusson, A simple bijection for the regions of the Shi arrangement of hyper-planes, Discrete Math. 204 (1999) 27-39.
7. n SIAM Journal on Discrete Mathematics 18 (2004) 397-417.
8. J.-C. Aval and N. Bergeron, Catalan paths and quasi-symmetric functions, Proc. Amer. Math. Soc. 131 (2003) 1053-1062.
9. D. Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. 36 (2003) 647-683.
10. M. Bestvina, Non-positively curved aspects of Artin groups of finite type. Geom. Topol. 3 (1999) 269-302.
11. P. Biane, Parking functions of types A and B, Electron. J. Combin. 9 (2002) 5 pp. (electronic).
12. T. Brady, A partial order on the symmetric group and new K(π 1)'s for the braid groups, Adv. Math. 161 (2001) 20-40.
13. T. Brady and C. Watt, K(π, 1)'s for Artin groups of finite type, Geom. Dedicata 94 (2002) 225-250.
14. F. Chapoton, S. Fomin, and A. Zelevinsky, Polytopal realizations of generalized associahedra. Canad. Math. Bull. 45 (2002) 537-566.
15. R. Charney, Geodesic automation and growth functions for Artin groups of finite type, Math. Ann. 301 (1995) 307-324.
16. R. Charney and M. W. Davis, Finite K(π,1l)s for Artin groups, in Prospects in Topology (Princeton, NJ, 1994), Princeton University Press, Princeton, 1995, pp. 110-124.
17. A. M. Cohen and D. B. Wales, Linearity of Artin groups of finite type, Israel J. Math. 131 (2002) 101-123.
18. A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.
19. _, A short survey of noncommutative geometry, J. Math. Phys. 41 (2000) 3832-3866.
20. N. Dershowitz and S. Zaks, Ordered trees and noncrossing partitions, Discrete Math. 62 (1986) 215-218.
21. P. H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980) 171-180.
22. _, Multichains, noncrossing partitions and trees, Discrete Math. 40 (1982) 171-179.
23. P. H. Edelman and R. Simion, Chains in the lattice of noncrossing partitions, Discrete Math. 126 (1994) 107-119.
24. R. Ehrenborg, On posets and Hopf algebras, Adv. Math. 119 (1996) 1-25.
25. S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977-1018.
26. J. Gilman, W. Menasco, and X.-S. Lin, eds., Knots, Braids, and Mapping Class Groups - Papers Dedicated to Joan S. Birman, American Mathematical Society, Providence, 2001.
27. I. Grossman and W. Magnus, Groups and Their Graphs, Random House, New York, 1964.
28. V. Jones and V. S. Sunder, Introduction to Subfactors, Cambridge University Press, Cambridge, 1997.
29. L. H. Kalikow, Symmetries in trees and parking functions, Discrete Math. 256 (2002) 719-741.
30. C. Kassel, M. Rosso, and V. Turaev, Quantum Groups and Knot Invariants, Société Mathématique de France, Paris, 1997.
31. 4 is linear, Invent. Math. 142 (2000) 451-486.
32. _, Braid groups are linear, Ann. of Math. 155(2002) 131-156.
33. G. Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 (1972) 333-350.
34. K. Murasugi and B. I. Kurpita, A Study of Braids, Kluwer, Dordrecht, 1999.
35. A. Nica and R. Speicher, A "Fourier transform" for multiplicative functions on non-crossing partitions, J. Algebraic Combin. 6 (1997) 141-160.
36. V. V. Prasolov and A. B. Sossinsky, Knots, Links, Braids and 3-Manifolds, American Mathematical Society, Providence, 1997.
37. H. Prodinger, A correspondence between ordered trees and noncrossing partitions, Discrete Math. 46 (1983) 205-206.
38. D. Rolfsen, New developments in the theory of Artin's braid groups, Topology Appl. 127 (2003) 77-90.
39. R. Simion, Noncrossing partitions, Discrete Math. 217 (2000) 367-409.
40. R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math. 98 (1991) 193-206.
41. R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994) 611-628.
42. _, Free probability theory and non-crossing partitions, Sém. Lothar. Combin. 39 (1997) 38 pp. (electronic).
43. _, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), x+88.
44. R. P. Stanley, Exercises on Catalan numbers and related numbers, available electronically at http://www-math.mit.edu/~rstan/ec/.
45. _, Parking functions and noncrossing partitions, Electron. J. Combin. 4 (1997) 14 pp. (electronic).
46. _, Enumerative Combinatorics, vol. 1, Cambridge University Press, Cambridge, 1997.
47. _, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999.
48. R. P. Stanley and I. Pitman, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (2002) 603-634.
49. D. Voiculescu, Free probability theory: Random matrices and von Neumann algebras, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), 2 vols., Birkhäuser, Basel, 1995, pp. 227-241.
50. _, Lectures on free probability theory, in Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Springer-Verlag, Berlin, 2000, pp. 279-349.
51. C. H. Yan, On the enumeration of generalized parking functions, Congr. Numer. 147 (2000) 201-209.