A060344 - OEIS

%I #20 Aug 04 2023 22:28:55

%S 2,6,20,52,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,

%T 196608,393216,786432,1572864,3145728,6291456,12582912,25165824,

%U 50331648,100663296,201326592,402653184,805306368,1610612736,3221225472

%N 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.

%D 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.

%D 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.

%H Harry J. Smith, <a href="/A060344/b060344.txt">Table of n, a(n) for n = 2..500</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).

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

%F G.f.: 2*x^2*(1 + x + 4*x^2 + 6*x^3 - 4*x^4)/(1 - 2*x). - _Colin Barker_, Mar 08 2012

%e 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.

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

%o (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

%Y Cf. A003945.

%K nonn,easy

%O 2,1

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

%E More terms from _Harry J. Smith_, Jul 04 2009