This is a related problem to the maximum determinant problem: http://www.indiana.edu/~maxdet/ The maximum determinant problem was the subject of a programming contest run by Lars Bakstrom in 2005: http://recmath.org/contest/Determinant/matrix.html [This contest was similar in spirit to the Al Zimmermann contests that used to be held regularly; I miss them greatly.] Recirculant matrices played a significant role in the results, and many tricks like the FFT trick were discussed. It's a great problem. On Sat, Aug 18, 2012 at 10:34 AM, W. Edwin Clark <wclark@mail.usf.edu>wrote:
On Sat, Aug 18, 2012 at 12:43 PM, Warren Smith <warren.wds@gmail.com> wrote:
What is the maximum |determinant| of an NxN circulant matrix whose defining first row consists entirely of +1's and -1's? (Related question: each entry lies between -1 and +1 inclusive?)
By changing slightly my (brute force) Maple program for http://oeis.org/A144926 for your question I get the following:
1, 1, [-1] 2, 0, 3, 4, [-1, -1, 1] 4, 16, [-1, -1, -1, 1] 5, 48, [-1, -1, -1, -1, 1] 6, 128, [-1, -1, -1, -1, -1, 1] 7, 512, [-1, -1, -1, 1, -1, 1, 1] 8, 2304, [-1, -1, -1, -1, 1, -1, 1, 1] 9, 6912, [-1, -1, -1, -1, -1, 1, -1, 1, 1] 10, 22528, [-1, -1, -1, -1, -1, -1, 1, -1, 1, 1] 11, 273408, [-1, -1, -1, -1, -1, 1, -1, -1, 1, 1, 1] 12, 2097152, [-1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1] 13, 14929920, [-1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1] 14, 50331648, [-1, -1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1] 15, 390905856, [-1, -1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, 1]
The sequence 1, 0, 4, 16, 48, 128, 512, 2304, 6912, 22528, 273408, 2097152, 14929920, 50331648, 390905856 is apparently not in the OEIS. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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