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Number of Deodhar elements in the finite Weyl group D_n.
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%I #3 Feb 27 2009 03:00:00

%S 2,5,14,48,167,575,1976,6791

%N Number of Deodhar elements in the finite Weyl group D_n.

%C The Deodhar elements are a subset of the fully commutative elements. If w is Deodhar, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w} and the Kazhdan-Lusztig basis element C'_w is the product of C'_{s_i}'s corresponding to any reduced expression for w.

%D S. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin., 13(2):111-136, 2001.

%D V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata, 36(1): 95-119, 1990.

%H S. C. Billey and B. C. Jones, <a href="http://www.arXiv.org/abs/math.CO/0612043">Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory</a>.

%e a(4)=48 because there are 48 fully commutative elements in D_4 and since the first non-Deodhar fully-commutative element does not appear until D_6, these are all of the Deodhar elements in D_4.

%Y Cf. A058094.

%K nonn

%O 1,1

%A Brant Jones (brant(AT)math.washington.edu), May 17 2007