A003432 -id:A003432 - cp's OEIS frontend

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

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

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

A108150 Number of different nonnegative values taken by the determinant of a real (0,1)-matrix of order n.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 10, 22, 46, 114, 294

Offset: 0

William P. Orrick, Jan 12 2006

For references and links see A089472.

A089472 is the main entry for this sequence. Cf. A003432, A013588, A051236.

a(1)=2, a(n) = (A089472(n) + 1) / 2 for n>1

a(0)=1 prepended by Alois P. Heinz, Mar 17 2019

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 16, 1, 0, 1, 6, 19, 40, 55, 32, 1, 0, 1, 7, 26, 66, 121, 144, 64, 1, 0, 1, 8, 34, 100, 221, 364, 377, 128, 1, 0, 1, 9, 43, 143, 364, 728, 1093, 987, 256, 1, 0, 1, 10, 53, 196, 560, 1288, 2380, 3280, 2584, 512, 1, 0

Offset: 0

N. J. A. Sloane, Jul 03 2015

Table 3.1 in Hopkins thesis is the same below the main diagonal. - F. Chapoton, Sep 04 2025

			The first few antidiagonals are:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  1,  0;
  1, 3,  4,  1,   0;
  1, 4,  8,  8,   1,   0;
  1, 5, 13, 21,  16,   1,  0;
  1, 6, 19, 40,  55,  32,  1, 0;
  1, 7, 26, 66, 121, 144, 64, 1, 0;
  ...
Square array starts:
  [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
  [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
  [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
  [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
  [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
  [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
  [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
  [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
  [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
  [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Noureddine Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G_r(Cⁿ), arXiv:hep-th/9808170, 1998. (Cf. Table 4).
E. Hopkins, Free Complexes over the Exterior Algebra with Small Homology, Ph.D. thesis, University of Nebraska, 2021. (Cf. Table 3.1.)
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
A(n,n) gives A274969.
Cf. A309896.
A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.

F:= proc(n) option remember;
      `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
    end:
A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 04 2015

F[n_] := F[n] = If[n<2, 1, Expand[F[n-1] - t*F[n-2]]]; A[n_, k_] := SeriesCoefficient[1/F[n+1], { t, 0, k}]; Table[A[d-k, k], {d, 0, 12}, {k, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

# The lower triangular array computed by F. Chapoton's formula:
from math import comb as binomial
def T(n: int, k: int) -> int:
    if k < 0:  return 0
    if k == 0: return 1
    if k == 1: return n
    return (binomial(n + 2*k, k - 2) * (n**2 + 3*n - 2*k + 2)) // (k * (k - 1))
for n in range(9): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Sep 06 2025

@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
def A(n, k): return F(n+1, 2*k)
print([A(n-k, k) for n in (0..11) for k in (0..n)]) # Peter Luschny, Aug 21 2019

Let F(n, k) = Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0. Then A(n, k) = F(n+1, 2*k). See [Shibukawa] and A309896. - Peter Luschny, Aug 21 2019

For n >= k >= 2, A(n, k) = binomial(n+2*k,k-2)*(n^2+3*n-2*k+2)/(k*(k-1)). - F. Chapoton, Sep 05 2025

More terms from Alois P. Heinz, Jul 04 2015

A373915 Determinants of symmetric real {0,1}-matrices of order 9 that only occur as positive values.

Original entry on oeis.org

89, 94, 95, 97, 98, 99, 101, 102, 105, 110, 116, 125, 144

Offset: 1

Hugo Pfoertner, Jun 23 2024

The non-occurrence of the corresponding negative determinants causes the difference A089472(9) - A118985(9) = 227 - 214 = 13.

A similar situation already occurs for 7 X 7 matrices, where the 3 determinant values -20, -24, -32 only occur with negative signs.

Hugo Pfoertner, Examples of 9 X 9 matrices with determinants = a(n).

Cf. A003432, A089472, A108150, A118985.

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.

A306837 Number of unimodular n X n matrices with elements {0, 1}.

Original entry on oeis.org

1, 6, 168, 20040, 9702720, 18480102480, 135491563468800, 3766962568171582080

Offset: 1

Steven E. Thornton, Mar 12 2019

An integer matrix is unimodular if its determinant is -1 or +1.

S. E. Thornton, Properties of the Bohemian family of n x n matrices with population {0, 1}, Characteristic Polynomial Database.

Number of different values taken by the determinant is in A089472.
Maximum determinant is in A003432.
A363862 gives equivalence classes up to row and column permutation.

a(7) and a(8) from Brendan McKay, Jun 25 2023

A373916 Numbers that are the determinant of real {0,1}-matrices of order 10, but not of symmetric such matrices.

Original entry on oeis.org

253, 268, 274, 294, 304

Offset: 1

Hugo Pfoertner, Jun 25 2024

This property also applies to the corresponding negative values.

William P. Orrick, Spectrum of the determinant function, (2010). The applicable part is "Spectrum for n=11" because of the representation for {-1,1}-matrices.
Hugo Pfoertner, Positions of the terms in a histogram of the accumulated occurrence counts.

Cf. A003432, A006125, A013588, A089472, A108150, A118985, A373915.

A215644 Full spectrum threshold for maximal determinant {+1, -1} matrices: largest order of submatrix for which the full spectrum of absolute determinant values occurs.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 4, 6, 6, 7, 6, 7, 7, 7, 8, 8, 8, 9, 8, 10

Offset: 1

Richard P. Brent and Judy-anne Osborn, Aug 18 2012

a(n) is the maximum of m(A) taken over all maximal determinant matrices A of order n, where m(A) is the maximum m such that the full spectrum of possible values (ignoring sign) occurs for the minors of order m of A.

			For n = 8 we have a(8) = 4 as a Hadamard matrix of order 8 has minors of order 4 with the full spectrum of values {0,8,16} (signs are ignored) but minors of order m > 4 do not have this property.

R. P. Brent, The Hadamard Maximal Determinant Problem
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv:1208.3819, 2012.
Index entries for sequences related to maximal determinants

Cf. A003432, A003433, A013588.

We calculated the first 21 terms of the sequence by an exhaustive computation of minors of known maximal determinant matrices as at August 2012.

A268408 Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.

Original entry on oeis.org

0, 1, 0, 4, 12, 56, 2, 1360, 2672, 320, 16, 350000, 431232, 107904, 12864, 3872, 320, 255036992, 234667968, 98251776, 19523136, 10633728, 1615552, 1182720, 163520, 127360, 13440

Offset: 1

R. J. Mathar, Feb 04 2016

The unit hypercube in dimension d has 2^d vertices, conveniently expressed by their Cartesian coordinates as binary vectors of length d of 0's and 1's. Hyper-tetrahedra (simplices) are defined by selecting a subset of 1+d of them. The (signed) volume V of a tetrahedron is the determinant of the d vectors of the edges divided by d!. (The volume may be zero if some edges in the tetrahedron are linearly dependent.) The triangle T(d,v) is a histogram of all A136465(d+1) tetrahedra classified by absolute (unsigned) volume V=v/d!.

The number of non-flat simplices (row sums without the leftmost column) are tabulated by Brandts et al. (Table 1, column beta_n). - R. J. Mathar, Feb 06 2016

			In d=2, 4 tetrahedra (triangles) are defined by taking subsets of d+1=3 vertices out of the 2^2=4 vertices of the unit square. Each of them has the same volume (area) 1/2!, so T(d=2,v=1)=4.
In d=3, 12 = T(d=3,v=0) tetrahedra with zero volume are defined by taking subsets of d+1=4 vertices out of the 2^3=8 vertices of the unit cube. These are the cases of taking any 4 vertices on a common face. (There are 6 faces and two different edge sets for each of them; one with edges along the cube's edges, and one with edges along the face diagonals.)
The triangle starts in row d=1 as follows:
0 1;
0  4;
12  56  2;
1360  2672  320  16 ;
350000  431232  107904  12864  3872  320;

J. Brandts, Sander Dijkhuis et al, There are only two nonobtuse binary triangulations of the unit n-cube, arXiv:1209.3875 and Comput. Geometry 46 (2013) 286

Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).

A123222 Expansion of -x * (x-1) * (3x^2-1) / (9x^4-8x^3+4x-1).

Original entry on oeis.org

1, 3, 9, 31, 109, 391, 1397, 4995, 17833, 63675, 227313, 811543, 2897269, 10343647, 36928061, 131837979, 470678161, 1680380979, 5999172633, 21417807055, 76464283837, 272987183095, 974598829637, 3479441311347, 12422046335161

Offset: 1

Roger L. Bagula, Oct 05 2006

Index entries for linear recurrences with constant coefficients, signature (4,0,-8,9).

Cf. A000952, A074872, A085340, A003432.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1))); // G. C. Greubel, Oct 12 2018

seq(coeff(series(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 13 2018

LinearRecurrence[{4,0,-8,9},{1,3,9,31},30] (* Harvey P. Dale, Jul 26 2018 *)

x='x+O('x^30); Vec(-x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1)) \\ G. C. Greubel, Oct 12 2018

From Colin Barker, Oct 19 2012: (Start)
a(n) = 4*a(n-1) -8*a(n-3) +9*a(n-4).
G.f.: -x*(x-1)*(3*x^2-1)/(9*x^4-8*x^3+4*x-1). (End)

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012

cp's OEIS Frontend

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

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A108150 Number of different nonnegative values taken by the determinant of a real (0,1)-matrix of order n.

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A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

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A373915 Determinants of symmetric real {0,1}-matrices of order 9 that only occur as positive values.

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

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Maple

Mathematica

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A306837 Number of unimodular n X n matrices with elements {0, 1}.

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A373916 Numbers that are the determinant of real {0,1}-matrices of order 10, but not of symmetric such matrices.

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A215644 Full spectrum threshold for maximal determinant {+1, -1} matrices: largest order of submatrix for which the full spectrum of absolute determinant values occurs.

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Links

Crossrefs

Extensions

A268408 Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.

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Author

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A123222 Expansion of -x * (x-1) * (3x^2-1) / (9x^4-8x^3+4x-1).