A061539 Number of signed permutations in B_n which correspond to smooth Schubert varieties. These permutations avoid the following patterns: (-2 -1) (1 2 -3) (1 -2 -3) (-1 2 -3) (2 -1 -3) (-2 1 -3) (3 -2 1) (2 -4 3 1) (-2 -4 3 1) (3412) (3 4 -1 2) (-3 4 1 2) (4 1 3 -2) (4 -1 3 -2) (4 2 3 1) (4 2 3 -1) (-4 2 3 1).

1, 2, 7, 28, 116, 490, 2094, 9014, 38988, 169184, 735846, 3205830, 13984076, 61057108, 266780436, 1166320956, 5101254296, 22319861332, 97685806958, 427635145446, 1872400460940, 8199602319764, 35912342632908, 157304824211156, 689096352589448, 3018916616772272

Offset

0,2

Comments

A signed permutation w corresponds to a matrix with exactly one nonzero entry in each row and column and that entry is either 1 or -1. A signed permutation avoids the pattern (1 2 -3) if no three rows and three columns gives a submatrix with diagonal entries 1 1 -1.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

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.

Formula

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

Example

a(2) = 7 because there are 8 signed permutations of two elements and there is exactly one bad pattern of length 2.

Prog

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

Crossrefs

Cf. A032351.

Sequence in context: A150646 A364145 A128611 * A232970 A116078 A150647

Adjacent sequences: A061536 A061537 A061538 * A061540 A061541 A061542

Keyword

nonn,easy

Author

Sara Billey, May 15 2001

Extensions

a(0)=1 prepended and a(10)-a(25) from Edward Richmond, Apr 05 2021

Status

approved