A060344 For n >= 2, let N_n denote the set of all unipotent upper-triangular real n X n matrices A such that for every k=1,2,...,n-1 the minor of A with rows 1,2,...,k and columns n-k+1,...,n is nonzero. a(n) is the number of connected components of N_n.

2, 6, 20, 52, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset

2,1

References

B. Shapiro, M. Shapiro and A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in SL_n/B, Internat. Math. Res. Notices, 1997, no. 10, pp. 469-493.

B. Shapiro, M. Shapiro and A. Vainshtein, Skew-symmetric vanishing lattices and intersections of Schubert cells. Internat. Math. Res. Notices, 1998, no. 11, pp. 563-588.

Links

Harry J. Smith, Table of n, a(n) for n = 2..500

Index entries for linear recurrences with constant coefficients, signature (2).

Formula

a(2)=2, a(3)=6, a(4)=20, a(5)=52; for n > 5, a(n) = 3 * 2^(n-1).

G.f.: 2*x^2*(1 + x + 4*x^2 + 6*x^3 - 4*x^4)/(1 - 2*x). - Colin Barker, Mar 08 2012

Example

a(2) = 2 because in this case the set N_n is topologically just the set of nonzero real numbers so the number of components is 2.

Mathematica

Join[{2, 6, 20, 52}, Table[3 2^(n-1), {n, 6, 40}]] (* Harvey P. Dale, Mar 09 2014 *)

Prog

(PARI) { for (n=2, 500, a=3*2^(n - 1); if (n==2, a=2); if (n==3, a=6); if (n==4, a=20); if (n==5, a=52); write("b060344.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

Crossrefs

Cf. A003945.

Sequence in context: A027558 A320910 A066397 * A363600 A347582 A045655

Adjacent sequences: A060341 A060342 A060343 * A060345 A060346 A060347

Keyword

nonn,easy

Author

Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001

Extensions

More terms from Harry J. Smith, Jul 04 2009

Status

approved