A224066 Number of smooth Schubert varieties of type C_n.

1, 2, 7, 28, 114, 472, 1988, 8480, 36474, 157720, 684404, 2976994, 12971206, 56587676, 247097170, 1079749976, 4720841314, 20649303934, 90353041092, 395459463960, 1731251197242, 7580521689750, 33197447406682, 145400339328566, 636901149067534, 2790082285204966

Offset: 0

Author

Sara Billey, Apr 02 2013

Keywords

• nonn
• easy

Comments

Characterized as the signed permutations avoiding the list of patterns: '((1 -2) (-2 -1 -3) (3 -2 1) (3 -2 -1) (-3 2 -1) (-3 -2 1) (-3 -2 -1)(-2 -4 3 1) (3 4 1 2) (3 4 -1 2) (-3 4 1 2) (-3 4 -1 2)(-3 -4 -1 -2) (4 -1 3 -2) (4 2 3 1) (4 2 3 -1) (-4 2 3 1))

Links

• S. C. 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.

Crossrefs

Cf. A061539.

Programs

PARI

seq(n)={Vec(((1-7*x+15*x^2-11*x^3-2*x^4+5*x^5)+(x-x^2-x^3+3*x^4-x^5)*sqrt(1-4*x + O(x^n)))/((1-x)^2*(1-6*x+8*x^2-4*x^3)))} \\ Andrew Howroyd, Apr 06 2021

Formula

G.f.: ((1-7*x+15*x^2-11*x^3-2*x^4+5*x^5)+(x-x^2-x^3+3*x^4-x^5)*sqrt(1-4*x))/((1-x)^2*(1-6*x+8*x^2-4*x^3)). - Edward Richmond, Apr 06 2021

Extensions

a(0)=1 prepended and a(11) and beyond added by Edward Richmond, Apr 05 2021