

A003432


Hadamard maximal determinant problem: largest determinant of a (real) {0,1}matrix of order n.
(Formerly M0720)


22



1, 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, 25515, 131072, 327680, 1114112, 3411968, 19531250, 56640625, 195312500
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OFFSET

0,4


COMMENTS

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^(n1)*a(n1). 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).
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 EhlichWojtas, and are therefore maximal. See the OrrickSolomon web site for further information. [Edited by William P. Orrick, Dec 20 2011]
The entry a(21) = 195312500 is now known to be correct. [Edited by Richard P. Brent, Aug 17 2021]


REFERENCES

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240246.
F. J. MacWilliams and N. J. A. Sloane, The Theory of ErrorCorrecting Codes, ElsevierNorth Holland, 1978, p. 54.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=0..21.
Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626630.
R. P. Brent, W. P. Orrick, J. Osborn, and P. Zimmermann, Maximal determinants and saturated Doptimal designs of orders 19 and 37, arXiv:1112.4160 [math.CO], 2011. [From William P. Orrick, Dec 20 2011]
Richard P. Brent and Judyanne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
V. Chasiotis, S. Kounias, and N. Farmakis, The Doptimal saturated designs of order 22, Discrete Mathematics 341 (2018), 380387. Corrigendum ibid 342 (2019), 2161.
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964), 123132.
H. Ehlich and K. Zeller, Binäre Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 2021.
J. Freixas and S. Kurz, On alpharoughly weighted games, arXiv preprint arXiv:1112.2861 [math.CO], 2011.
Matthew Hudelson, Victor Klee and David Larman, Largest jsimplices in dcubes: 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), 519598.
J. Huttenhain and C. Ikenmeyer, Binary Determinantal Complexity, arXiv:1410.8202 [cs.CC], 20142015.
Yongbin Li, Junwei Zi, Yan Liu and Xiaojun Zhang, A note of values of minors for Hadamard matrices, arXiv:1905.04662 [math.CO], 2019.
William P. Orrick, The maximal {1, 1}determinant of order 15, arXiv:math/0401179 [math.CO], 2004.
William P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1, arXiv:math/0311292 [math.CO], 2003.
William P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)
William P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem, arXiv:math/0304410 [math.CO], 2003.
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
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427434. Math. Rev. 8,128g.
Luke Zeng, Shawn Xin, Avadesian Xu, Thomas Pang, Tim Yang and Maolin Zheng, Seele's New AntiASIC Consensus Algorithm with Emphasis on Matrix Computation, arXiv:1905.04565 [cs.CR], 2019.
Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181211.
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants


EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 32*x^7 + 56*x^8 + ...
One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:
1 0 0 1 1 0
0 0 1 1 1 1
1 1 1 0 0 1
0 1 0 1 0 1
0 1 0 0 1 1
0 1 1 1 1 0


CROSSREFS

A003433(n) = 2^(n1)*a(n1). Cf. A013588, A036297, A051752.
Sequence in context: A278119 A118998 A276410 * A179332 A081938 A129500
Adjacent sequences: A003429 A003430 A003431 * A003433 A003434 A003435


KEYWORD

nonn,hard,more,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

a(18)a(20) added by William P. Orrick, Dec 20 2011
a(21) added by Richard P. Brent, Aug 16 2021


STATUS

approved



