Generalized noncrossing partitions and combinatorics of Coxeter groups.

*(English)*Zbl 1191.05095
Mem. Am. Math. Soc. 202, No. 949, x, 159 p. (2009).

The memoir under review is a refinement of the author’s PhD thesis, written at Cornell University in 2006. Besides the description of the author’s research, this nice text also provides a concise and unified account of the present state of the field indicated in the title. In addition, the author suggests some open problems for future research and raises several conjectures, many of which have since then been object of research by other authors.

The central object of this memoir is a generalization of the lattice of noncrossing partitions: the poset \(\text{NC}^{(k)}(W)\), defined for each finite real Coxeter group \(W\), and each extended integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. When \(k=1\), this poset coincides with the poset \(\text{NC}(W)\) of generalized noncrossing partitions introduced by T. Brady and C. Watt [Geom. Dedicata 94, 225–250 (2002; Zbl 1053.20034)], and, when \(W\) is the symmetric group, it corresponds to the poset of classical \(k\)-divisible noncrossing partitions first studied by P. H. Edelman [Discrete Math. 31, 171–180 (1980; Zbl 0443.05011)].

The author presents a solid study of both the structural and the enumerative properties of \(\text{NC}^{(k)}(W)\) and, in the case that \(W\) is a classical Coxeter group of type \(A\) or \(B\), shows that \(\text{NC}^{(k)}(W)\) is isomorphic to a poset of noncrossing set partitions in which each block has size divisible by \(k\). The author also identifies many enumerative features that the poset \(\text{NC}^{(k)}(W)\) shares with the generalized nonnesting partitions of C. A. Athanasiadis [Bull. Lond. Math. Soc. 36, No. 3, 294–302 (2004; Zbl 1068.20038)], and with the generalized cluster complexes of S. Fomin and N. Reading [Int. Math. Res. Not. 2005, No. 44, 2709–2757 (2005; Zbl 1117.52017)], and presents several conjectures relating these three families of objects. These families currently include what is known as the “Fuss-Catalan objects”, whose central enumerative feature is a generalized Coxeter-Catalan number, known as Fuss-Catalan number.

Here are the chapter headings, followed by some comments:

1. Introduction.

The author sketches his motivation and gives an outline of the thesis.

2. Coxeter Groups and Noncrossing Partitions.

This chapter starts with a quick introduction to the theory of finite Coxeter systems and root systems, following the approach in [J. E. Humphreys, Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press (1990; Zbl 0725.20028)]. A combinatorial approach to classical Coxeter systems via the theory of reduced S-words and the weak order is given in Section 2.3. In the following two sections, the author describes the more recent approach to this subject using the theory of reduced T-words and the absolute order. The lattice \(\text{NC}(W)\) of noncrossing partitions corresponding to the finite Coxeter group \(W\) is defined in Section 2.6, and some of its properties are discussed. Finally, in the last section, the notions of degrees and exponents for the Coxeter group \(W\) are introduced, again following Humphreys, and the generalized Coxeter-Catalan number is defined.

3. \(k\)-Divisible Noncrossing Partitions.

This chapter is the heart of the memoir. Here, the author introduces and studies the poset \(\text{NC}^{(k)}(W)\) for each finite real Coxeter group \(W\) and each extended positive integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. Structural and enumerative properties of \(\text{NC}^{(k)}(W)\) are explored in sections 3.4 through 3.7. In general, it is shown that \(\text{NC}^{(k)}(W)\) is a graded join-semilattice whose elements are counted by a generalized “Fuss-Catalan number”, which can be given by a closed formula in terms of the degrees of basic invariants of \(W\). The author also shows that this poset is locally self-dual, and, he computes the number of multichains in \(\text{NC}^{(k)}(W)\), encoded by the zeta polynomial. The order complex of the poset is shown to be shellable, and therefore Cohen-Macaulay, and, its homotopy type is computed. Finally, it is shown that the rank numbers of \(\text{NC}^{(k)}(W)\) are polynomials in \(k\) with nonzero rational coefficients alternating in sign. This defines a new family of polynomials, called “Fuss-Narayana”, associated with the pair \((W,k)\), and some properties of these polynomials are described.

4. The Classical Types.

In the first two sections of this chapter, the author introduces the classical theory of noncrossing partitions as initiated by G. Kreweras [Discrete Math. 1, 333–350 (1972; Zbl 0231.05014)]. Afterwards, he explores the results from chapter 3 in the context of classical groups, and proves some case-by-case results. In particular, combinatorial realizations of the posets \(\text{NC}^{(k)}(A_{n-1})\) and \(\text{NC}^{(k)}(B_n)\) as posets of set partitions in which each block has size divisible by \(k\) are given. Krattenthaler has recently given a similar characterization of the poset \(\text{NC}^{(k)}(D_n)\) using annular noncrossing partitions.

5. Fuss-Catalan Combinatorics.

In the final chapter of this memoir, the author describes the other two subjects that, together with the poset of \(k\)-divisible noncrossing partitions, currently comprise the Fuss-Catalan combinatorics of Coxeter groups. The first one is the theory of the generalized nonnesting partitions of Athanasiadis, and the second one is the theory of generalized cluster complexes of Fomin and Reading. Several coincidences between the three families of Fuss-Catalan objects are described, and the author makes several conjectures related to these families.

The central object of this memoir is a generalization of the lattice of noncrossing partitions: the poset \(\text{NC}^{(k)}(W)\), defined for each finite real Coxeter group \(W\), and each extended integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. When \(k=1\), this poset coincides with the poset \(\text{NC}(W)\) of generalized noncrossing partitions introduced by T. Brady and C. Watt [Geom. Dedicata 94, 225–250 (2002; Zbl 1053.20034)], and, when \(W\) is the symmetric group, it corresponds to the poset of classical \(k\)-divisible noncrossing partitions first studied by P. H. Edelman [Discrete Math. 31, 171–180 (1980; Zbl 0443.05011)].

The author presents a solid study of both the structural and the enumerative properties of \(\text{NC}^{(k)}(W)\) and, in the case that \(W\) is a classical Coxeter group of type \(A\) or \(B\), shows that \(\text{NC}^{(k)}(W)\) is isomorphic to a poset of noncrossing set partitions in which each block has size divisible by \(k\). The author also identifies many enumerative features that the poset \(\text{NC}^{(k)}(W)\) shares with the generalized nonnesting partitions of C. A. Athanasiadis [Bull. Lond. Math. Soc. 36, No. 3, 294–302 (2004; Zbl 1068.20038)], and with the generalized cluster complexes of S. Fomin and N. Reading [Int. Math. Res. Not. 2005, No. 44, 2709–2757 (2005; Zbl 1117.52017)], and presents several conjectures relating these three families of objects. These families currently include what is known as the “Fuss-Catalan objects”, whose central enumerative feature is a generalized Coxeter-Catalan number, known as Fuss-Catalan number.

Here are the chapter headings, followed by some comments:

1. Introduction.

The author sketches his motivation and gives an outline of the thesis.

2. Coxeter Groups and Noncrossing Partitions.

This chapter starts with a quick introduction to the theory of finite Coxeter systems and root systems, following the approach in [J. E. Humphreys, Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press (1990; Zbl 0725.20028)]. A combinatorial approach to classical Coxeter systems via the theory of reduced S-words and the weak order is given in Section 2.3. In the following two sections, the author describes the more recent approach to this subject using the theory of reduced T-words and the absolute order. The lattice \(\text{NC}(W)\) of noncrossing partitions corresponding to the finite Coxeter group \(W\) is defined in Section 2.6, and some of its properties are discussed. Finally, in the last section, the notions of degrees and exponents for the Coxeter group \(W\) are introduced, again following Humphreys, and the generalized Coxeter-Catalan number is defined.

3. \(k\)-Divisible Noncrossing Partitions.

This chapter is the heart of the memoir. Here, the author introduces and studies the poset \(\text{NC}^{(k)}(W)\) for each finite real Coxeter group \(W\) and each extended positive integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. Structural and enumerative properties of \(\text{NC}^{(k)}(W)\) are explored in sections 3.4 through 3.7. In general, it is shown that \(\text{NC}^{(k)}(W)\) is a graded join-semilattice whose elements are counted by a generalized “Fuss-Catalan number”, which can be given by a closed formula in terms of the degrees of basic invariants of \(W\). The author also shows that this poset is locally self-dual, and, he computes the number of multichains in \(\text{NC}^{(k)}(W)\), encoded by the zeta polynomial. The order complex of the poset is shown to be shellable, and therefore Cohen-Macaulay, and, its homotopy type is computed. Finally, it is shown that the rank numbers of \(\text{NC}^{(k)}(W)\) are polynomials in \(k\) with nonzero rational coefficients alternating in sign. This defines a new family of polynomials, called “Fuss-Narayana”, associated with the pair \((W,k)\), and some properties of these polynomials are described.

4. The Classical Types.

In the first two sections of this chapter, the author introduces the classical theory of noncrossing partitions as initiated by G. Kreweras [Discrete Math. 1, 333–350 (1972; Zbl 0231.05014)]. Afterwards, he explores the results from chapter 3 in the context of classical groups, and proves some case-by-case results. In particular, combinatorial realizations of the posets \(\text{NC}^{(k)}(A_{n-1})\) and \(\text{NC}^{(k)}(B_n)\) as posets of set partitions in which each block has size divisible by \(k\) are given. Krattenthaler has recently given a similar characterization of the poset \(\text{NC}^{(k)}(D_n)\) using annular noncrossing partitions.

5. Fuss-Catalan Combinatorics.

In the final chapter of this memoir, the author describes the other two subjects that, together with the poset of \(k\)-divisible noncrossing partitions, currently comprise the Fuss-Catalan combinatorics of Coxeter groups. The first one is the theory of the generalized nonnesting partitions of Athanasiadis, and the second one is the theory of generalized cluster complexes of Fomin and Reading. Several coincidences between the three families of Fuss-Catalan objects are described, and the author makes several conjectures related to these families.

Reviewer: Ricardo Mamede (Coimbra)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05A17 | Combinatorial aspects of partitions of integers |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

05E18 | Group actions on combinatorial structures |

06A06 | Partial orders, general |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

05A18 | Partitions of sets |

##### Keywords:

noncrossing partition; Coxeter group; Coxeter element; Catalan number; Fuss-Catalan number; nonnesting partition; cluster complex##### Citations:

Zbl 1053.20034; Zbl 0443.05011; Zbl 1068.20038; Zbl 1117.52017; Zbl 0725.20028; Zbl 0231.05014
PDF
BibTeX
XML
Cite

\textit{D. Armstrong}, Generalized noncrossing partitions and combinatorics of Coxeter groups. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1191.05095)

**OpenURL**

##### References:

[1] | D. Armstrong, Braid groups, clusters, and free probability: an outline from the AIM workshop, January 2005; available at www.aimath.org/WWN/braidgroups/braidgroups.pdf. |

[2] | D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, PhD thesis, Cornell University, 2006. |

[3] | Christos A. Athanasiadis, Deformations of Coxeter hyperplane arrangements and their characteristic polynomials, Arrangements-Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 1-26. · Zbl 0976.32016 |

[4] | Christos A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc. 36 (2004), no. 3, 294-302. · Zbl 1068.20038 |

[5] | Christos A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), no. 1, 179-196 (electronic). · Zbl 1079.20057 |

[6] | Christos A. Athanasiadis, On noncrossing and nonnesting partitions for classical reflection groups, Electron. J. Combin. 5 (1998), Research Paper 42, 16 pp. (electronic). · Zbl 0898.05004 |

[7] | Christos A. Athanasiadis, On some enumerative aspects of generalized associahedra, European J. Combin. 28 (2007), no. 4, 1208-1215. · Zbl 1117.52013 |

[8] | Christos A. Athanasiadis, Thomas Brady, Jon McCammond, and Colum Watt, \(h\)-vectors of generalized associahedra and noncrossing partitions, Int. Math. Res. Not. , posted on (2006), Art. ID 69705, 28. · Zbl 1112.20032 |

[9] | Christos A. Athanasiadis, Thomas Brady, and Colum Watt, Shellability of noncrossing partition lattices, Proc. Amer. Math. Soc. 135 (2007), no. 4, 939-949 (electronic). · Zbl 1171.05053 |

[10] | Christos A. Athanasiadis and Victor Reiner, Noncrossing partitions for the group \(D_n\), SIAM J. Discrete Math. 18 (2004), no. 2, 397-417 (electronic). · Zbl 1085.06001 |

[11] | Christos A. Athanasiadis and Eleni Tzanaki, On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements, J. Algebraic Combin. 23 (2006), no. 4, 355-375. · Zbl 1107.20024 |

[12] | Christos A. Athanasiadis and Eleni Tzanaki, Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes, Israel J. Math. 167 (2008), 177-191. · Zbl 1200.05252 |

[13] | Kenneth Baclawski, Cohen-Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), no. 4, 293-305. · Zbl 0504.06005 |

[14] | Hélène Barcelo and Edwin Ihrig, Lattices of parabolic subgroups in connection with hyperplane arrangements, J. Algebraic Combin. 9 (1999), no. 1, 5-24. · Zbl 0938.20023 |

[15] | Hélène Barcelo and Alain Goupil, Combinatorial aspects of the Poincaré polynomial associated with a reflection group, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 21-44. · Zbl 0812.05070 |

[16] | H.W. Becker, The general theory of rhyme, Bull. Amer. Math. Soc. 52 (1946), 415. |

[17] | H. W. Becker, Rooks and Rhymes, Math. Mag. 22 (1948), no. 1, 23-26. |

[18] | H.W. Becker, Planar rhyme schemes, Bull. Amer. Math. Soc. 58 (1952), 39. |

[19] | David Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 647-683 (English, with English and French summaries). · Zbl 1064.20039 |

[20] | D. Bessis, Topology of complex reflection groups, preprint, arXiv.org/math.GT/0411645. · Zbl 0134.22805 |

[21] | David Bessis and Ruth Corran, Non-crossing partitions of type \((e,e,r)\), Adv. Math. 202 (2006), no. 1, 1-49. · Zbl 1128.20024 |

[22] | David Bessis, François Digne, and Jean Michel, Springer theory in braid groups and the Birman-Ko-Lee monoid, Pacific J. Math. 205 (2002), no. 2, 287-309. · Zbl 1056.20023 |

[23] | D. Bessis and V. Reiner, Cyclic sieving of noncrossing partitions for complex reflection groups, preprint, arXiv.org/math.CO/0701792. · Zbl 1268.20041 |

[24] | P. Biane, Free probability and combinatorics, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 765-774. · Zbl 1003.46035 |

[25] | Philippe Biane, Some properties of crossings and partitions, Discrete Math. 175 (1997), no. 1-3, 41-53. · Zbl 0892.05006 |

[26] | Philippe Biane, Frederick Goodman, and Alexandru Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2263-2303 (electronic). · Zbl 1031.46075 |

[27] | Louis J. Billera and Bernd Sturmfels, Fiber polytopes, Ann. of Math. (2) 135 (1992), no. 3, 527-549. · Zbl 0762.52003 |

[28] | Louis J. Billera and Bernd Sturmfels, Iterated fiber polytopes, Mathematika 41 (1994), no. 2, 348-363. · Zbl 0819.52010 |

[29] | Joan Birman, Ki Hyoung Ko, and Sang Jin Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math. 139 (1998), no. 2, 322-353. · Zbl 0937.20016 |

[30] | Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159-183. · Zbl 0441.06002 |

[31] | A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 1819-1872. · Zbl 0851.52016 |

[32] | Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. · Zbl 1110.05001 |

[33] | Nicolas Bourbaki, Elements of the history of mathematics, Springer-Verlag, Berlin, 1994. Translated from the 1984 French original by John Meldrum. · Zbl 0803.01002 |

[34] | N. Brady, J. Crisp, A. Kaul and J. McCammond, Factoring isometries, poset completions and Artin groups of affine type, in preparation. |

[35] | Thomas Brady, A partial order on the symmetric group and new \(K(\pi ,1)\)’s for the braid groups, Adv. Math. 161 (2001), no. 1, 20-40. · Zbl 1011.20040 |

[36] | Thomas Brady, Artin groups of finite type with three generators, Michigan Math. J. 47 (2000), no. 2, 313-324. · Zbl 0996.20022 |

[37] | Thomas Brady and Colum Watt, A partial order on the orthogonal group, Comm. Algebra 30 (2002), no. 8, 3749-3754. · Zbl 1018.20040 |

[38] | Thomas Brady and Colum Watt, \(K(\pi ,1)\)’s for Artin groups of finite type, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 225-250. · Zbl 1053.20034 |

[39] | T. Brady and C. Watt, Lattices in finite real reflection groups, preprint, arXiv.org/math.CO/0501502. · Zbl 1187.20051 |

[40] | Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. · Zbl 0715.20017 |

[41] | W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. · Zbl 0064.25105 |

[42] | D. Callan and L. Smiley, Noncrossing partitions under reflection and rotation, preprint, arXiv.org/math.CO/0510447. |

[43] | R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1-59. · Zbl 0254.17005 |

[44] | A. Cayley, On the partition of a polygon, Proc. London Math. Soc. 22(1) (1891), 237-262. · JFM 23.0541.01 |

[45] | Paola Cellini and Paolo Papi, Ad-nilpotent ideals of a Borel subalgebra. II, J. Algebra 258 (2002), no. 1, 112-121. Special issue in celebration of Claudio Procesi’s 60th birthday. · Zbl 1033.17008 |

[46] | Frédéric Chapoton, Enumerative properties of generalized associahedra, Sém. Lothar. Combin. 51 (2004/05), Art. B51b, 16 pp. (electronic). · Zbl 1160.05342 |

[47] | F. Chapoton, Sur le nombre de réflexions pleines dans les groupes de Coxeter finis, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 585-596 (French, with English and French summaries). · Zbl 1150.20023 |

[48] | Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537-566. Dedicated to Robert V. Moody. · Zbl 1018.52007 |

[49] | Liang Chen, Robert V. Moody, and Jiří Patera, Non-crystallographic root systems, Quasicrystals and discrete geometry (Toronto, ON, 1995) Fields Inst. Monogr., vol. 10, Amer. Math. Soc., Providence, RI, 1998, pp. 135-178. · Zbl 0916.20026 |

[50] | William Y. C. Chen, Eva Y. P. Deng, Rosena R. X. Du, Richard P. Stanley, and Catherine H. Yan, Crossings and nestings of matchings and partitions, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1555-1575. · Zbl 1108.05012 |

[51] | Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379-436. · Zbl 0359.20029 |

[52] | H. S. M. Coxeter, Regular polytopes, 3rd ed., Dover Publications, Inc., New York, 1973. · Zbl 0031.06502 |

[53] | H.S.M. Coxeter, The complete enumeration of finite groups of the form \(R_i^2=(R_iR_j)^{k_{ij}}=1\), J. London Math. Soc. 10 (1935), 21-25. · Zbl 0010.34202 |

[54] | H. S. M. Coxeter, The product of the generators of a finite group generated by reflections, Duke Math. J. 18 (1951), 765-782. · Zbl 0044.25603 |

[55] | N. Dershowitz, Ordered trees and non-crossing partitions, Discrete Math. 31 (1986), 215-218. · Zbl 0646.05004 |

[56] | Dragomir Ž. Djoković, On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups, Proc. Amer. Math. Soc. 80 (1980), no. 1, 181-184. · Zbl 0436.22008 |

[57] | Paul H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980), no. 2, 171-180. · Zbl 0443.05011 |

[58] | Paul H. Edelman, Multichains, noncrossing partitions and trees, Discrete Math. 40 (1982), no. 2-3, 171-179. · Zbl 0496.05007 |

[59] | P. H. Edelman and V. Reiner, Free arrangements and rhombic tilings, Discrete Comput. Geom. 15 (1996), no. 3, 307-340. · Zbl 0853.52013 |

[60] | Sen-Peng Eu and Tung-Shan Fu, The cyclic sieving phenomenon for faces of generalized cluster complexes, Adv. in Appl. Math. 40 (2008), no. 3, 350-376. · Zbl 1147.05014 |

[61] | Sergey Fomin and Nathan Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005), 2709-2757. · Zbl 1117.52017 |

[62] | Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 63-131. · Zbl 1147.52005 |

[63] | Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 (electronic). · Zbl 1021.16017 |

[64] | Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121. · Zbl 1054.17024 |

[65] | Sergey Fomin and Andrei Zelevinsky, Cluster algebras: notes for the CDM-03 conference, Current developments in mathematics, 2003, Int. Press, Somerville, MA, 2003, pp. 1-34. · Zbl 1119.05108 |

[66] | Sergey Fomin and Andrei Zelevinsky, \(Y\)-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018. · Zbl 1057.52003 |

[67] | N. Fuss, Solio quæstionis, quot modis polygonum n laterum in polyga m laterum, per diagonales resolvi quæat, Nova acta academiæscientarium Petropolitanæ9 (1791), 243-251. |

[68] | A. M. Garsia and M. Haiman, A remarkable \(q,t\)-Catalan sequence and \(q\)-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191-244. · Zbl 0853.05008 |

[69] | F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235-254. · Zbl 0194.03303 |

[70] | I. M. Gel\(^{\prime}\)fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. · Zbl 0827.14036 |

[71] | Iain Gordon, On the quotient ring by diagonal invariants, Invent. Math. 153 (2003), no. 3, 503-518. · Zbl 1039.20019 |

[72] | Curtis Greene, Posets of shuffles, J. Combin. Theory Ser. A 47 (1988), no. 2, 191-206. · Zbl 0659.06001 |

[73] | S. Griffeth, Finite dimensional modules for rational Cherednik algebras, preprint, arXiv.org/math.CO/0612733. |

[74] | H.T. Hall, Meanders in a Cayley graph, preprint, arXiv.org/math.CO/0606170. |

[75] | Mark D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17-76. · Zbl 0803.13010 |

[76] | Peter Hilton and Jean Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (1991), no. 2, 64-75. · Zbl 0767.05010 |

[77] | Christophe Hohlweg and Carsten E. M. C. Lange, Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), no. 4, 517-543. · Zbl 1125.52011 |

[78] | C. Hohlweg, C. Lange and H. Thomas, Permutahedra and Generalized Associahedra, preprint, arXiv.org/math.CO/0709.4241. · Zbl 1233.20035 |

[79] | James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028 |

[80] | Richard Kane, Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5, Springer-Verlag, New York, 2001. · Zbl 0986.20038 |

[81] | T.P. Kirkman, On the \(k\)-partitions of the \(r\)-gon and \(r\)-ace, Phil. Trans. R. Soc. London 147 (1857), 217-272. |

[82] | Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032. · Zbl 0099.25603 |

[83] | Anisse Kasraoui and Jiang Zeng, Distribution of crossings, nestings and alignments of two edges in matchings and partitions, Electron. J. Combin. 13 (2006), no. 1, Research Paper 33, 12 pp. (electronic). · Zbl 1096.05006 |

[84] | Christian Krattenthaler, The \(F\)-triangle of the generalised cluster complex, Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, Berlin, 2006, pp. 93-126. · Zbl 1122.05096 |

[85] | C. Krattenthaler, The \(M\)-triangle of generalised non-crossing partitions for the types \(E_7\) and \(E_8\), Séminaire Lotharingien Combin. 54 (2006), Article B54l, 34 pages. |

[86] | C. Krattenthaler, Non-crossing partitions on an annulus, in preparation. |

[87] | C. Krattenthaler and T. Müller, Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions, preprint, arXiv.org/math.CO/0704.0199. · Zbl 1228.05306 |

[88] | G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333-350 (French). · Zbl 0231.05014 |

[89] | Carl W. Lee, The associahedron and triangulations of the \(n\)-gon, European J. Combin. 10 (1989), no. 6, 551-560. · Zbl 0682.52004 |

[90] | Jean-Louis Loday, Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), no. 3, 267-278. · Zbl 1059.52017 |

[91] | J. McCammond, An introduction to Garside structures, preprint; available at www.math.ucsb.edu/\~mccammon. · Zbl 0732.20034 |

[92] | Jon McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006), no. 7, 598-610. · Zbl 1179.05015 |

[93] | C. Montenegro, The fixed point non-crossing partition lattices, manuscript, 1993. |

[94] | Th. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Amer. Math. Soc. 54 (1948), 352-360. · Zbl 0032.24607 |

[95] | Alexandru Nica and Roland Speicher, A “Fourier transform” for multiplicative functions on non-crossing partitions, J. Algebraic Combin. 6 (1997), no. 2, 141-160. · Zbl 0872.06005 |

[96] | Alexandru Nica and Roland Speicher, On the multiplication of free \(N\)-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), no. 4, 799-837. · Zbl 0856.46035 |

[97] | Peter Orlik and Louis Solomon, Unitary reflection groups and cohomology, Invent. Math. 59 (1980), no. 1, 77-94. · Zbl 0452.20050 |

[98] | Alexander Postnikov and Richard P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 544-597. In memory of Gian-Carlo Rota. · Zbl 0962.05004 |

[99] | Alexander Postnikov, Symmetries of Gromov-Witten invariants, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 251-258. · Zbl 1041.14024 |

[100] | Józef H. Przytycki and Adam S. Sikora, Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers, J. Combin. Theory Ser. A 92 (2000), no. 1, 68-76. · Zbl 0959.05004 |

[101] | D. Ranjan, Counting triangulations of a convex polygon, webpage, www.math.nmsu.edu/hist_projects/pascalII.pdf |

[102] | Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313-353. · Zbl 1106.20033 |

[103] | Nathan Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5931-5958. · Zbl 1189.05022 |

[104] | N. Reading and D. Speyer, Cambrian fans, preprint, arXiv.org/math.CO/0606210. · Zbl 1213.20038 |

[105] | Victor Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195-222. · Zbl 0892.06001 |

[106] | V. Reiner, D. Stanton, and D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004), no. 1, 17-50. · Zbl 1052.05068 |

[107] | Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. · Zbl 0964.05070 |

[108] | G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304. · Zbl 0055.14305 |

[109] | Jian-Yi Shi, The number of \(\oplus \)-sign types, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 189, 93-105. · Zbl 0889.20024 |

[110] | Rodica Simion, A type-B associahedron, Adv. in Appl. Math. 30 (2003), no. 1-2, 2-25. Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001). · Zbl 1047.52006 |

[111] | 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). · Zbl 0959.05009 |

[112] | Rodica Simion and Frank W. Schmidt, Restricted permutations, European J. Combin. 6 (1985), no. 4, 383-406. · Zbl 0615.05002 |

[113] | Rodica Simion and Richard P. Stanley, Flag-symmetry of the poset of shuffles and a local action of the symmetric group, Discrete Math. 204 (1999), no. 1-3, 369-396. · Zbl 0931.06004 |

[114] | Rodica Simion and Daniel Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math. 98 (1991), no. 3, 193-206. · Zbl 0760.05004 |

[115] | Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57-64. · Zbl 0117.27104 |

[116] | Eric N. Sommers, \(B\)-stable ideals in the nilradical of a Borel subalgebra, Canad. Math. Bull. 48 (2005), no. 3, 460-472. · Zbl 1139.17303 |

[117] | Roland Speicher, Free probability theory and non-crossing partitions, Sém. Lothar. Combin. 39 (1997), Art. B39c, 38 pp. (electronic). · Zbl 0887.46036 |

[118] | Roland Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), no. 4, 611-628. · Zbl 0791.06010 |

[119] | T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159-198. · Zbl 0287.20043 |

[120] | Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. · Zbl 0838.13008 |

[121] | Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. · Zbl 0889.05001 |

[122] | Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. · Zbl 0928.05001 |

[123] | Richard P. Stanley, Flag-symmetric and locally rank-symmetric partially ordered sets, Electron. J. Combin. 3 (1996), no. 2, Research Paper 6, approx. 22 pp. (electronic). The Foata Festschrift. · Zbl 0857.05091 |

[124] | Richard P. Stanley, Parking functions and noncrossing partitions, Electron. J. Combin. 4 (1997), no. 2, Research Paper 20, approx. 14 pp. (electronic). The Wilf Festschrift (Philadelphia, PA, 1996). · Zbl 0883.06001 |

[125] | Richard P. Stanley, Recent progress in algebraic combinatorics, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 55-68. Mathematical challenges of the 21st century (Los Angeles, CA, 2000). · Zbl 1119.14002 |

[126] | Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132-161. · Zbl 0496.06001 |

[127] | James Dillon Stasheff, Homotopy associativity of \(H\)-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312. · Zbl 0114.39402 |

[128] | J. Stembridge, coxeter and posets, Maple packages for working with finite Coxeter groups and partially-ordered sets; available at http://math.lsa.umich.edu/\~jrs. |

[129] | Hugh Thomas, An analogue of distributivity for ungraded lattices, Order 23 (2006), no. 2-3, 249-269. · Zbl 1134.06003 |

[130] | J. Tits, Groupes et géométries de Coxeter, IHES, 1961. |

[131] | Eleni Tzanaki, Faces of generalized cluster complexes and noncrossing partitions, SIAM J. Discrete Math. 22 (2008), no. 1, 15-30. · Zbl 1222.20029 |

[132] | Eleni Tzanaki, Polygon dissections and some generalizations of cluster complexes, J. Combin. Theory Ser. A 113 (2006), no. 6, 1189-1198. · Zbl 1105.52009 |

[133] | Masahiko Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math. 157 (2004), no. 2, 449-454. · Zbl 1113.52039 |

[134] | Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. · Zbl 0296.50010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.