A224243 Number of smooth, equivalently rationally smooth, Schubert varieties of type D_n.

4, 22, 108, 490, 2164, 9474, 41374, 180614, 788676, 3445462, 15059202, 65847946, 288033326, 1260313930, 5516051890, 24147542122, 105729680608, 463006798298, 2027839420598, 8882324416302, 38909820194506, 170461077652718, 746826223566214, 3272185833672630

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..25.

Sara Billey, Pattern avoidance and rational smoothness of Schubert varieties, Advances in Math, vol. 139 (1998) pp. 141-156.

E. Richmond and W. Slofstra, Staircase diagrams and enumeration of smooth Schubert varieties, arXiv:1510.06060 [math.CO], 2015; J. Combin. Ser. A, Vol 150 (2017) pp. 328-376.

Formula

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

Prog

(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

Crossrefs

Cf. A061539, A224240.

Sequence in context: A184510 A184701 A001436 * A024420 A220457 A108840

Adjacent sequences: A224240 A224241 A224242 * A224244 A224245 A224246

Keyword

nonn,easy

Author

Sara Billey, Apr 01 2013

Extensions

Offset corrected and terms a(10) and beyond from Edward Richmond, Apr 05 2021

Status

approved