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%I #34 Apr 07 2021 12:22:27
%S 4,22,108,490,2164,9474,41374,180614,788676,3445462,15059202,65847946,
%T 288033326,1260313930,5516051890,24147542122,105729680608,
%U 463006798298,2027839420598,8882324416302,38909820194506,170461077652718,746826223566214,3272185833672630
%N Number of smooth, equivalently rationally smooth, Schubert varieties of type D_n.
%C Also, the number of signed permutations w in the Type D signed permutations whose initial interval [id,w] in Bruhat order is rank symmetric. Equivalently, the Kazhdan-Lusztig polynomial P_id,w(q) = 1. Characterized by pattern avoidance.
%H Sara Billey, <a href="https://doi.org/10.1006/aima.1998.1744">Pattern avoidance and rational smoothness of Schubert varieties</a>, Advances in Math, vol. 139 (1998) pp. 141-156.
%H E. Richmond and W. Slofstra, <a href="https://arxiv.org/abs/1510.06060">Staircase diagrams and enumeration of smooth Schubert varieties</a>, arXiv:1510.06060 [math.CO], 2015; J. Combin. Ser. A, Vol 150 (2017) pp. 328-376.
%F G.f.: 4*x^2 + x*((-4 + 19*x + 8*x^2 - 30*x^3 + 16*x^4)*(1-x) + (4 - 15*x^1 + 11*x^2 - 2*x^4)*sqrt(1-4*x))/((1-x)*(1 - 6*x + 8*x^2 - 4*x^3)). - _Edward Richmond_, Apr 05 2021
%o (PARI) seq(n)={Vec(4*x+((-4+19*x+8*x^2-30*x^3+16*x^4)*(1 - x)+(4 -15*x^1 + 11*x^2 -2*x^4)*sqrt(1-4*x + O(x*x^n)))/((1-x)*(1-6*x+8*x^2-4*x^3)))} \\ _Andrew Howroyd_, Apr 06 2021
%Y Cf. A061539, A224240.
%K nonn,easy
%O 2,1
%A _Sara Billey_, Apr 01 2013
%E Offset corrected and terms a(10) and beyond from _Edward Richmond_, Apr 05 2021
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