A051752 -id:A051752 - cp's OEIS frontend

A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

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

Offset: 0

N. J. A. Sloane

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).
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 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]

			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

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
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).

Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1-simplices 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), 626-630.
R. P. Brent, W. P. Orrick, J. Osborn, and P. Zimmermann, Maximal determinants and saturated D-optimal designs of orders 19 and 37, arXiv:1112.4160 [math.CO], 2011. [From William P. Orrick, Dec 20 2011]
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 18.
V. Chasiotis, S. Kounias, and N. Farmakis, The D-optimal saturated designs of order 22, Discrete Mathematics 341 (2018), 380-387. Corrigendum ibid 342 (2019), 2161.
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964), 123-132.
H. Ehlich and K. Zeller, Binäre Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.
J. Freixas and S. Kurz, On alpha-roughly weighted games, arXiv preprint arXiv:1112.2861 [math.CO], 2011.
Matthew Hudelson, Victor Klee and David Larman, 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.
J. Huttenhain and C. Ikenmeyer, Binary Determinantal Complexity, arXiv:1410.8202 [cs.CC], 2014-2015.
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), 427-434. Math. Rev. 8,128g.
Luke Zeng, Shawn Xin, Avadesian Xu, Thomas Pang, Tim Yang and Maolin Zheng, Seele's New Anti-ASIC 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), 181-211.
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.

a(18)-a(20) added by William P. Orrick, Dec 20 2011

a(21) added by Richard P. Brent, Aug 16 2021

A086264 Number of real {0,1} n X n matrices having determinant=1.

Original entry on oeis.org

1, 1, 3, 84, 10020, 4851360, 9240051240, 67745781734400, 1883481284085791040

Offset: 0

Hugo Pfoertner, Oct 05 2003

Hugo Pfoertner, Determinants of (0,1)-matrices. FORTRAN program.
Minfeng Wang, C++ program
Eric Weisstein's World of Mathematics, (0,1)-Matrix.
Index entries for sequences related to binary matrices

Cf. A003024, A003432, A051752, A055165.

Cf. A046747.

a[n_] := Module[{M, iter, cnt = 0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[Det[M] == 1, cnt++], Evaluate[Sequence @@ iter]]; cnt];
Do[Print[n, " ", a[n]], {n, 1, 4}] (* Jean-François Alcover, Dec 09 2018 *)

a(0)=1 prepended by Alois P. Heinz, Jun 18 2022
a(7) from Minfeng Wang, Feb 09 2023
a(8) from Minfeng Wang, Apr 26 2024

A089478 Triangle T(n,k) read by rows, where T(n,k) = number of times the determinant of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= A003432(n).

Original entry on oeis.org

0, 1, 1, 1, 10, 3, 338, 84, 3, 42976, 10020, 1200, 60, 21040112, 4851360, 1213920, 144720, 43560, 3600, 39882864736, 9240051240, 3868663680, 768723480, 418703040, 63612360, 46569600, 6438600, 5014800, 529200, 292604283435872

Offset: 0

Hugo Pfoertner, Nov 04 2003

The first 4 rows were provided by Wouter Meeussen.

			a(4) = T(2,1) = 3 because there are 3 different (0,1)-matrices with determinant=1:
  ((1,0),(0,1)), ((1,1),(0,1)), ((1,0),(1,1)).
Triangle T(n,k) begins:
         0,       1;
         1,       1;
        10,       3;
       338,      84,       3;
     42976,   10020,    1200,     60;
  21040112, 4851360, 1213920, 144720, 43560, 3600;
  ...

Minfeng Wang, Table of n, a(n) for n = 0..118

Cf. T(n,0) = A046747(n), T(n,1) = A086264(n), T(n,A003432(n)) = A051752(n).
The n-th row of the table contains A089472(n) nonzero entries.
Cf. A089479.

Fortran
```
c See link in A086264.
```

Edited by Alois P. Heinz, Dec 20 2023

A119004 Number of n X n real symmetric (0,1)-matrices having maximal determinant (=A119002(n)).

Original entry on oeis.org

1, 1, 1, 18, 160, 900, 2520, 36960, 393120, 15573600

Offset: 1

Giovanni Resta, May 08 2006

Index entries for sequences related to maximal determinants

Cf. A051752, A119002, A119005, A119006.

a(8)-a(10) from Max Alekseyev, Jun 17 2025

A188895 Number of n X n (real) {-1,1}-matrices having determinant A003433(n).

Original entry on oeis.org

1, 4, 96, 384, 30720, 7372800, 4335206400, 2477260800, 25684239974400, 7191587192832000

Offset: 0

Eric W. Weisstein, Apr 19 2011

Minfeng Wang, C++ program to calculate a(2) .. a(5)
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem
Eric Weisstein's World of Mathematics, (-1, 1)-Matrix

Cf. A003433, A051752.

a(n) = 2^(2n+1) * A051752(n) for n>=2. - Hugo Pfoertner and Minfeng Wang, Jan 22 2023

Offset changed to 0 by Hugo Pfoertner, Jan 23 2023

a(5)-a(9) from Minfeng Wang, Jan 22 2023

A052655 a(2) = 6, otherwise a(n) = n*n!.

Original entry on oeis.org

0, 1, 6, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = number of real non-singular (0,1)-matrices of order n having maximal permanent = A000255(n). Proof: [W. Edwin Clark and Richard Brualdi] The maximum permanent is per A where A has all 1's except for n-1 0's on the main diagonal. By Corollary 4.4 in the Brualdi et al. reference for n >= 4 any n X n (0,1)-matrix B with per B = per A can be obtained from A by permuting rows and columns. Since there are n ways to place the single 1 on the main diagonal and then n! ways to permute the distinct rows, a(n) = n*n! if n >=4. Direct computation shows this also holds for n = 1 and 3. - W. Edwin Clark, Nov 15 2003

			a(2)=6 because there are 6 (0,1)-matrices with nonzero determinant having permanent=1. See example in A089482. The (0,1)-matrix with maximal permanent=2 ((1,1),(1,1)) has det=0.

Richard A. Brualdi, John L. Goldwasser, T. S. Michael, Maximum permanents of matrices of zeros and ones, J. Combin. Theory Ser. A 47 (1988), 207-245.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 602

Cf. A000255. A089480 gives occurrence counts for permanents of non-singular (0, 1)-matrices, A051752 number of (0, 1)-matrices with maximal determinant A003432.

Essentially the same as A001563.

spec := [S,{S=Prod(Z,Union(Z,Prod(Sequence(Z),Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0,1,6},Table[n*n!,{n,3,20}]] (* Harvey P. Dale, Apr 20 2012 *)

E.g.f.: x*(-2*x^2+x^3+x+1)/(-1+x)^2.

cp's OEIS Frontend

A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

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A086264 Number of real {0,1} n X n matrices having determinant=1.

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A089478 Triangle T(n,k) read by rows, where T(n,k) = number of times the determinant of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= A003432(n).

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A119004 Number of n X n real symmetric (0,1)-matrices having maximal determinant (=A119002(n)).

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A188895 Number of n X n (real) {-1,1}-matrices having determinant A003433(n).

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A052655 a(2) = 6, otherwise a(n) = n*n!.

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