cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A224240 Number of rationally smooth Schubert varieties of type B_n.

Original entry on oeis.org

1, 8, 34, 142, 596, 2530, 10842, 46766, 202594, 880210
Offset: 1

Views

Author

Sara Billey, Apr 01 2013

Keywords

Comments

Also, the number of signed permutations w in the hyperoctahedral group 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.

References

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

Crossrefs

Cf. A061539.

A224066 Number of smooth Schubert varieties of type C_n.

Original entry on oeis.org

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

Views

Author

Sara Billey, Apr 02 2013

Keywords

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

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

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

Original entry on oeis.org

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

Views

Author

Sara Billey, Apr 01 2013

Keywords

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

Crossrefs

Cf. A061539, A224240.

Programs

  • 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

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

Extensions

Offset corrected and terms a(10) and beyond from Edward Richmond, Apr 05 2021
Showing 1-3 of 3 results.

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