Revision History for A060344
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| A060344
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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.
(history;
published version)
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#20 by Ray Chandler at Fri Aug 04 22:28:55 EDT 2023
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#19 by Ray Chandler at Fri Aug 04 22:28:51 EDT 2023
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<a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
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#18 by Ray Chandler at Thu Jun 29 12:17:49 EDT 2023
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#17 by Ray Chandler at Thu Jun 29 12:17:46 EDT 2023
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<a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
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#16 by Ray Chandler at Thu Jun 29 12:17:45 EDT 2023
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<a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
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#15 by Susanna Cuyler at Tue Jul 03 21:19:12 EDT 2018
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#14 by Jon E. Schoenfield at Tue Jul 03 20:26:08 EDT 2018
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#13 by Jon E. Schoenfield at Tue Jul 03 20:26:05 EDT 2018
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| NAME
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For n >= 2 Let, 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.
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| REFERENCES
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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.
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| FORMULA
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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
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| PROG
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(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); ) } [From _); ) } \\ _Harry J. Smith_, Jul 04 2009]
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#12 by Harvey P. Dale at Sun Mar 09 11:00:28 EDT 2014
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#11 by Harvey P. Dale at Sun Mar 09 11:00:23 EDT 2014
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| MATHEMATICA
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Join[{2, 6, 20, 52}, Table[3 2^(n-1), {n, 6, 40}]] (* Harvey P. Dale, Mar 09 2014 *)
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