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On the number of reduced decompositions of elements of Coxeter groups. (English) Zbl 0587.20002
Let r(w) denote the number of reduced decompositions of the element w of a Coxeter group W. The author gives a formula for r(w) in terms of symmetric functions when W is the symmetric group $$S_ n$$ (Weyl group of type A). This formula is quite explicit in many cases, e.g. if $$w_ 0$$ is the element of maximal length in $$S_ n$$, then $$r(w_ 0)$$ is equal to the number of standard Young tableaux of the (staircase) shape (n-1,n- 2,...,1). When W is the hyperoctahedral group (Weyl group of type B) the author formulates some conjectures for r(w) in analogy to the $$S_ n$$ case in terms of shifted standard tableaux. The situation for other Weyl groups remains unclear.
Reviewer: T.Jozefiak

##### MSC:
 20B30 Symmetric groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05A15 Exact enumeration problems, generating functions 20F05 Generators, relations, and presentations of groups
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##### References:
 [1] Berge, C., Principles of combinatorics, (1971), Academic Press New York · Zbl 0227.05002 [2] A. Björner, The weak ordering of Coxeter groups, Order, to appear. [3] Björner, A., Orderings of Coxeter groups, (), 175-195 [4] Björner, A.; Wachs, M., Bruhat order of coexter groups and shellability, Advances in math, 43, 87-100, (1982) · Zbl 0481.06002 [5] Bourbaki, N., Groups et algèbres de Lie, Éléments de mathématique, fasc. XXXIV, (1968), Hermann Paris, Chaps. 4, 5, and 6 · Zbl 0186.33001 [6] P. H. Edelman, A partial order on the regions of ℝ^n dissected by hyperplanes, preprint. · Zbl 0555.06003 [7] Edelman, P.; Greene, C., Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the Bruhat order of Sn, () · Zbl 0562.05008 [8] J. E. Goodman and R. Pollack, Semispaces of configurations, cell complexes of arrangements, J. Combin. Theory, Ser. A, to appear. · Zbl 0551.05002 [9] Guilbaud, G.Th.; Rosenstiehl, P., (), 72-100 [10] Humphreys, J.E., Introduction to Lie algebras and representation theory, (1972), Springer-Verlag New York · Zbl 0254.17004 [11] Lascoux, A.; Schützenberger, M.-P., (), 629-633, Serie 1 [12] Macdonald, I.G., Symmetric functions and Hall polynomials, (1979), Oxford University Press Oxford · Zbl 0487.20007 [13] Proctor, R., Shifted plane partitions of trapezoidal shape, Proc. American math. soc, 89, 553-559, (1983) · Zbl 0525.05007 [14] Stanley, R., Ordered structures and partitions, Memoirs of the American math. soc., no. 119, (1972) · Zbl 0246.05007 [15] Stanley, R., Theory and application of plane partitions, Parts 1 and 2, studies in applied math, 50, 167-188, (1971), 259-279 · Zbl 0225.05011 [16] Stanley, R., Weyl groups, the hard Lefschetz theorem, and the sperner property, SIAM. J. algebraic and discrete methods, 1, 168-184, (1980) · Zbl 0502.05004 [17] Yanagimoto, T.; Okamoto, M., Partial orderings of permutations and monotonicity of a rank correlation statistic, Ann. institute statistical math, 21, 489-506, (1969) · Zbl 0208.44704
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