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Enumeration

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  1. R. Adin, F. Brenti, Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 27 (2001), 210–224. [243]

    Article  MathSciNet  Google Scholar 

  2. H. Barcelo, A. Goupil, Non-broken circuits of reflection groups and factorization in D n. Israel J. Math. 91 (1995), 285–306. [242]

    MathSciNet  MATH  Google Scholar 

  3. S. Billey, M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443–482. [232, 244]

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Billey, T. K. Lam, Vexillary elements in the hyperoctahedral group, J. Algebraic Combin. 8 (1998), 139–152. [243]

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Björner, F. Brenti, Affine permutations of type A, Electron. J. Combin. 3 (1996), no. 2, #R18. [242, 293, 294]

    Google Scholar 

  6. A. Björner, M. LasVergnas, B. Sturmfels, N. White, G. M. Ziegler, Oriented Matroids, Cambridge University Press, Cambridge, 1993. [243, 299, 306]

    MATH  Google Scholar 

  7. A. Björner, M. Wachs, Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308 (1988), 1–37. [64, 242]

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Bott, An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84 (1956), 251–281. [242]

    MATH  MathSciNet  Google Scholar 

  9. M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions. Ramanujan J. 1 (1997), 101–111. [242]

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Brenti, q-Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), 417–441. [242, 243]

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Charney, M. Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), 373–378. [242]

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Chevalley, Sur certains groupes simples, Tohoku Math. J. 7 (1955), 14–66. [242]

    MATH  MathSciNet  Google Scholar 

  13. R. J. Clarke, D. Foata, Eulerian calculus. I. Univariable statistics, European J. Combin. 15 (1994), 345–362. [243]

    Article  MathSciNet  MATH  Google Scholar 

  14. R. J. Clarke, D. Foata, Eulerian calculus. II. An extension of Han’s fundamental transformation, European J. Combin. 16 (1995), 221–252. [243]

    Article  MathSciNet  MATH  Google Scholar 

  15. R. J. Clarke, D. Foata, Eulerian calculus. III. The ubiquitous Cauchy formula, European J. Combin. 16 (1995), 329–355. [243]

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Dolgachev, V. Lunts, A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra, to appear. [243]

    Google Scholar 

  17. M. Dyer, On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (2001), 2591–2595. [242]

    Article  MATH  MathSciNet  Google Scholar 

  18. P. H. Edelman, C. Greene, Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the weak Bruhat order of S n, Combinatorics and Algebra (Boulder, 1983), Contemp. Math. 34, American Mathematical Society, Providence, RI, 1984, pp. 155–162. [243]

    Google Scholar 

  19. P. H. Edelman, C. Greene, Balanced tableaux, Adv. Math. 63 (1987), 42–99. [243]

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Eriksson, K. Eriksson, Affine Weyl groups as infinite permutations. Electron. J. Combin. 5 (1998), no. 1, #R18, 32 pp. [208, 242, 294]

    Google Scholar 

  21. C. K. Fan, Schubert varieties and short braidedness, Transform. Groups 3 (1998), 51–56. [244]

    Article  MATH  MathSciNet  Google Scholar 

  22. C. K. Fan, J. R. Stembridge, Nilpotent orbits and commutative elements, J. Algebra 196 (1997), 490–498. [244]

    Article  MathSciNet  MATH  Google Scholar 

  23. D. Foata, G. N. Han, Calcul basique des permutations signées. I. Longueur et nombre d’inversions, Adv. Applied Math. 18 (1997), 489–509. [243]

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Fomin, A. Kirillov, Reduced words and plane partitions. J. Algebraic Combin. 6 (1997), 311–319. [242]

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Fomin, R. P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), 196–207. [242]

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), 79–113. [232, 242, 244]

    Article  MATH  MathSciNet  Google Scholar 

  27. W. Kraśkiewicz, Reduced decompositions in hyperoctahedral groups. C. R. Acad. Sci. Paris, Ser. I Math. 309 (1989), 903–907. [244]

    MathSciNet  MATH  Google Scholar 

  28. T. K. Lam, B and D analogues of stable Schubert polynomials and related insertion algorithms, Ph.D. Thesis, Massachusetts Institute of Technology, 1994. [244]

    Google Scholar 

  29. A. Lascoux, M.-P. Schützenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux. C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629–633. [243]

    MATH  Google Scholar 

  30. G. I. Lehrer, On the Poincaré series associated with Coxeter group actions on the complements of hyperplanes, J. London Math. Soc. 36 (1987), 275–294. [242]

    MATH  MathSciNet  Google Scholar 

  31. I. G. Macdonald, Notes on Schubert Polynomials, Publ. du LACIM 6, University du Québec, Montréal, 1991. [234, 242]

    Google Scholar 

  32. T. Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 15, Springer-Verlag, Berlin Heidelberg, 1988. [243]

    MATH  Google Scholar 

  33. A. Ram, Standard Young tableaux for finite root systems, preprint. [243]

    Google Scholar 

  34. V. Reiner, Quotients of Coxeter complexes and P-partitions, Mem. Amer. Math. Soc. 95 (1992), no. 460, vi+134 pp. [243]

    Google Scholar 

  35. V. Reiner, Signed posets, J. Combin. Theory Ser. A 62 (1993), 324–360. [243]

    Article  MATH  MathSciNet  Google Scholar 

  36. V. Reiner, Signed permutation statistics, European J. Combin. 14 (1993), 553–567. [243]

    Article  MATH  MathSciNet  Google Scholar 

  37. V. Reiner, Upper binomial posets and signed permutation statistics, European J. Combin. 14 (1993), 581–588. [243]

    Article  MATH  MathSciNet  Google Scholar 

  38. V. Reiner, Signed permutation statistics and cycle-type, European J. Combin. 14 (1993), 569–579. [243]

    Article  MATH  MathSciNet  Google Scholar 

  39. V. Reiner, The distribution of descent and length in a Coxeter group, Electron. J. Combin. 2 (1995), R25. [242]

    MATH  MathSciNet  Google Scholar 

  40. V. Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), 195–222. [243]

    Article  MATH  MathSciNet  Google Scholar 

  41. V. Reiner, G. Ziegler, Coxeter associahedra, Mathematika 41 (1994), 364–393. [243]

    Article  MathSciNet  MATH  Google Scholar 

  42. J. P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math. Studies, No. 70, Princeton University Press, Princeton, NJ, 1971, pp. 77–169. [242]

    Google Scholar 

  43. L. Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376–393. [242]

    Article  MATH  MathSciNet  Google Scholar 

  44. R. P. Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combin. Theory Ser. A 20 (1976), 336–356. [210, 242]

    Article  MATH  MathSciNet  Google Scholar 

  45. R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), 359–372. [234, 242, 243, 244]

    MATH  MathSciNet  Google Scholar 

  46. R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc. Nr. 80 (1968). [242]

    Google Scholar 

  47. E. Steingrimsson, Permutation statistics of indexed permutations, European. J. Combin. 15 (1994), 187–205. [243]

    Article  MATH  MathSciNet  Google Scholar 

  48. J. R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. 99(1992), 307–320. [243]

    Article  MATH  MathSciNet  Google Scholar 

  49. J. R. Stembridge, On the action of a Weyl group on the cohomology of its associated toric variety, Adv. Math. 106 (1994), 244–301. [243]

    Article  MATH  MathSciNet  Google Scholar 

  50. J. R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), 353–385. [244]

    MATH  MathSciNet  Google Scholar 

  51. J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), 1285–1332. [244]

    Article  MATH  MathSciNet  Google Scholar 

  52. J. R. Stembridge, The enumeration of fully commutative elements of Coxeter groups, J. Algebraic Combin. 7 (1998), 291–320. [244]

    Article  MATH  MathSciNet  Google Scholar 

  53. L. Tan, On the distinguished coset representatives of the parabolic subgroups in finite Coxeter groups, Commun. Algebra 22 (1994), 1049–1061. [242]

    MATH  Google Scholar 

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(2005). Enumeration. In: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol 231. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27596-7_7

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