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A003432
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Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.
(Formerly M0720)
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16
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1, 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, 25515, 131072, 327680, 1114112, 3411968, 19531250, 56640625
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OFFSET
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0,4
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COMMENTS
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The entries are restricted to 0 and 1; the determinant is computed in the field of real numbers.
Suppose M = (m(i,j)) is an n X n matrix of real numbers. Let
a(n) = max det M subject to m(i,j) = 0 or 1 [this sequence],
g(n) = max det M subject to m(i,j) = -1 or 1 [A003433],
h(n) = max det M subject to m(i,j) = -1, 0 or 1 [A003433],
F(n) = max det M subject to 0 <= m(i,j) <= 1 [this sequence],
G(n) = max det M subject to -1 <= m(i,j) <= 1 [A003433].
Then a(n) = F(n), g(n) = h(n) = G(n), g(n) = 2^(n-1)*a(n-1). Thus all five problems are equivalent.
Hadamard proved that a(n) <= 2^(-n)*(n+1)^((n+1)/2), with equality if and only if a Hadamard matrix of order n+1 exists. Equivalently, g(n) <= n^(n/2), with equality if and only if a Hadamard matrix of order n exists. It is believed that a Hadamard matrix of order n exists if and only if n = 1, 2 or a multiple of 4 (see A036297).
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REFERENCES
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J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.
H. Ehlich, Determinantenabschaetzungen fuer binaere Matrizen, Math. Z. 83 (1964), 123-132.
H. Ehlich and K. Zeller, Binaere Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.
J. Freixas and S. Kurz, On alpha-roughly weighted games, Arxiv preprint arXiv:1112.2861, 2011
J. Hadamard, R\'{e}solution d'une question relative aux d\'{e}terminants, Bull. des Sciences Math. 2 (1893), 240-246.
Hudelson, Matthew; Klee, Victor and Larman, David, Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem. Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 519-598.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 54.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.
C. Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.
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LINKS
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Table of n, a(n) for n=0..20.
R. P. Brent, W. P. Orrick, J. Osborn, and P. Zimmermann, Maximal determinants and saturated D-optimal designs of orders 19 and 37.
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, Arxiv preprint arXiv:1208.3819, 2012.
W. P. Orrick, The maximal {-1, 1}-determinant of order 15.
W. P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1
W. P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)
W. P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Hadamard's maximum determinant problem.
Eric Weisstein's World of Mathematics, (0, 1)-Matrix
Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants
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EXAMPLE
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One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:
100110
001111
111001
010101
010011
011110
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CROSSREFS
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A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.
Sequence in context: A105180 A094206 A118998 * A179332 A081938 A129500
Adjacent sequences: A003429 A003430 A003431 * A003433 A003434 A003435
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KEYWORD
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nonn,hard,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Added a(18)-a(20) and Brent, Orrick, Osborn, Zimmermann reference.
We have a(21) = 195312500?, a(22) = 662671875?, and a(36) = 1200757082375992968. Furthermore, starting with a(23), many constructions are known that attain the upper bounds of Hadamard, Barba, and Ehlich-Wojtas, and are therefore maximal. See the Orrick-Solomon web site for further information.
Edited by Will Orrick, 20 December 2011.
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STATUS
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approved
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