A096201 -id:A096201 - cp's OEIS frontend

A007299 Number of Hadamard matrices of order 4n.

1, 1, 1, 1, 5, 3, 60, 487, 13710027

Offset: 0

More precisely, number of inequivalent Hadamard matrices of order n if two matrices are considered equivalent if one can be obtained from the other by permuting rows, permuting columns and multiplying rows or columns by -1.
The Hadamard conjecture is that a(n) > 0 for all n >= 0. - Charles R Greathouse IV, Oct 08 2012
From Bernard Schott, Apr 24 2022: (Start)
A brief historical overview based on the article "La conjecture de Hadamard" (see link):
1893 - J. Hadamard proposes his conjecture: a Hadamard matrix of order 4k exists for every positive integer k (see link).
As of 2000, there were five multiples of 4 less than or equal to 1000 for which no Hadamard matrix of that order was known: 428, 668, 716, 764 and 892.
2005 - Hadi Kharaghani and Behruz Tayfeh-Rezaie publish their construction of a Hadamard matrix of order 428 (see link).
2007 - D. Z. Djoković publishes "Hadamard matrices of order 764 exist" and constructs 2 such matrices (see link).
As of today, there remain 12 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964. (End)
By private email, Felix A. Pahl informs that a Hadamard matrix of order 1004 was constructed in 2013 (see link Djoković, Golubitsky, Kotsireas); so 1004 is deleted from the last comment. - Bernard Schott, Jan 29 2023

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 1073, 2002.

V. Álvarez, J. A. Armario, M. D. Frau, and F. Gudiel, The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}, Linear Algebra and its Applications, 2011.
F. J. Aragon Artacho, J. M. Borwein, and M. K. Tam, Douglas-Rachford Feasibility Methods for Matrix Completion Problems, March 2014.
Dragomir Z. Djoković, Hadamard matrices of order 764 exist, arXiv:math/0703312 [math.CO], 2007.
Dragomir Z. Djoković, Oleg Golubitsky, and Ilias S. Kotsireas, Some new orders of Hadamard and skew-Hadamard matrices, arXiv:1301.3671 [math.CO], 2013.
Shalom Eliahou, La conjecture de Hadamard (I), Images des Mathématiques, CNRS, 2020.
Shalom Eliahou, La conjecture de Hadamard (II), Images des Mathématiques, CNRS, 2020.
Jacques Salomon Hadamard, Sur le module maximum que puisse atteindre un déterminant, C. R. Acad. Sci. Paris 116 (1893) 1500-1501 (link from Gallica).
Hadi Kharaghani, Home Page
Hadi Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, J. Combin. Designs 21 (2013) no. 5, 212-221. [DOI]
Hadi Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, Journal of Combinatorial Designs, Volume 13, Issue 6, November 2005, pp. 435-440 (First published: 13 December 2004).
H. Kharaghani and B. Tayfeh-Rezaie, On the classification of Hadamard matrices of order 32, J. Combin. Des., 18 (2010), 328-336.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, 2012.
H. Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory (1986), A 43, 98-102.
H. Kimura, New Hadamard matrix of order 24, Graphs Combin. (1989), 5, 235-242.
H. Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math. (1994), 128, 257-268.
H. Kimura, Classification of Hadamard matrices of order 28, Discrete Math. (1994), 133, 171-180.
W. P. Orrick, Switching operations for Hadamard matrices, arXiv:math/0507515 [math.CO], 2005-2007. (Gives lower bounds for a(8) and a(9))
N. J. A. Sloane, Tables of Hadamard matrices
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Warren D. Smith, C program for generating Hadamard matrices of various orders
Edward Spence, Classification of Hadamard matrices of order 24 and 28, Discrete Math. 140 (1995), no. 1-3, 185-243.
Eric Weisstein's World of Mathematics, Hadamard Matrix
Index entries for sequences related to Hadamard matrices

Cf. A019442, A096201, A036297, A048615, A048616, A003432, A048885.

a(8) from the H. Kharaghani and B. Tayfeh-Rezaie paper. - N. J. A. Sloane, Feb 11 2012

A353052 Number of inequivalent {-1,1} matrices of order n, up to permutation of rows and/or columns, multiplication of rows and/or columns by -1, and transposition.

Original entry on oeis.org

1, 2, 3, 10, 30, 242, 4386

Offset: 1

Nathaniel Johnston, Apr 20 2022

The equivalence operations described in the title are commonly used when discussing Hadamard matrices, for example (see A096201). They are natural when considering norms of these matrices or properties that can be inferred from their singular values, since they do not change singular values. See A352099 for the version of this sequence that does not consider transposition as part of the equivalence relation.

Since the row and column multiplication operations can be used to force the first row and column to consist only of ones, 2^[(n-1)^2] is an upper bound on this sequence. A lower bound is 2^[n*(n-2)] / (n!)^2.

			When n = 3, there are 3 inequivalent matrices, so a(3) = 3:
  1 1 1       1  1  1       1  1  1
  1 1 1       1  1 -1       1 -1 -1
  1 1 1  and  1 -1 -1  and  1 -1 -1
All other 3-by-3 matrices with entries in {-1,1} can be converted into one of these three matrices by permutating rows and/or columns, multiplying some rows and/or columns by -1, and potentially transposing the matrix.

John Holbrook, Nathaniel Johnston, and Jean-Pierre Schoch, Real Schur norms and Hadamard matrices, arXiv:2206.02863 [math.CO], 2022.
Nathaniel Johnston, All inequivalent matrices of size 6-by-6 or less

Cf. A111368, A352099.

a(7) from Nathaniel Johnston, May 05 2022

cp's OEIS Frontend

A007299 Number of Hadamard matrices of order 4n.

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A096204 Maximum permanent of a Hadamard matrix of order 4n.

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A096205 Minimum (absolute value of) permanent of a Hadamard matrix of order 4n.

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A096206 Number of different values taken by the absolute value of the permanent of a Hadamard matrix of order 4n.

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A353052 Number of inequivalent {-1,1} matrices of order n, up to permutation of rows and/or columns, multiplication of rows and/or columns by -1, and transposition.

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A206704 Number of Hadamard matrices of order 4n that are equivalent to their transpose.

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A206705 Number of Hadamard matrices of order n that are equivalent to their transpose.

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