A085000 -id:A085000 - cp's OEIS frontend

A351148 a(n) is the maximal determinant of a symmetric n X n matrix using the integers 1 to n*(n+1)/2.

1, 5, 112, 7113, 745285, 90536348

Offset: 1

Hugo Pfoertner at the suggestion of Markus Sigg, Feb 15 2022

			a(2) = 5:
  [2, 1;
   1, 3]
.
a(3) = 112:
  [1, 5, 4;
   5, 3, 6;
   4, 6, 2]
.
a(4) = 7113:
  [ 3, 10,  4,  8;
   10,  5,  1,  6;
    4,  1,  7,  9;
    8,  6,  9,  2]
.
a(5) = 745285:
  [10, 12,  1,  4, 13;
   12,  6,  5, 15,  3;
    1,  5,  9, 11, 14;
    4, 15, 11,  7,  2;
   13,  3, 14,  2,  8]
.
a(6) = 90536348:
  [18,  1,  7, 14,  6, 15;
    1, 13,  5, 17,  8, 20;
    7,  5, 12,  2, 21, 16;
   14, 17,  2, 11, 19,  3;
    6,  8, 21, 19,  9,  4;
   15, 20, 16,  3,  4, 10]

Cf. A000217, A085000, A351147, A351609.

A088745 Infinite array read by antidiagonals: for n>=1 let T_n = upper left n X n matrix. Then T_1 = (1), T_n has elements 1..n^2, contains T_{n-1} as n-1 X n-1 upper left submatrix and abs(det(T_n)) is a maximized.

Original entry on oeis.org

1, 3, 4, 8, 2, 6, 11, 7, 9, 15, 24, 16, 5, 12, 17, 26, 18, 13, 14, 25, 35, 48, 36, 22, 10, 23, 27, 37

Offset: 1

Paul D. Hanna, Oct 14 2003

Matrices with elements 1..n^2 that are to have maximum absolute determinant under the following construction. Start with a(1,1)=1, build successive n X n matrices by fixing previous (n-1) X (n-1) matrix and adding matrix elements (n-1)^2+1..n^2 on lower and right border of matrix. Determinants of upper left n X n matrices are: {1, -10, 205, -6300, 276363, -15615642, ...}.
The definition is incomplete since it does not say what to do if there are several possibilities for the new border. - N. J. A. Sloane, Oct 18 2003
Terms computed by Hugo Pfoertner (see link). If we start with either 2 X 2 matrices [1,3][4,2] or [1,4][3,2], initially there seems to be a unique solution for the subsequent enhanced matrices.

			The 7 X 7 subarray is:
[ 1, 3, 8, 11, 24, 26, 48]
[ 4, 2, 7, 16, 18, 36, 39]
[ 6, 9, 5, 13, 22, 30, 40]
[15, 12, 14, 10, 21, 34, 42]
[17, 25, 23, 20, 19, 31, 47]
[35, 27, 29, 33, 32, 28, 45]
[37, 43, 46, 41, 49, 44, 38]

Hugo Pfoertner, Construction of maximal determinants. FORTRAN program.

Cf. A088746 (determinants), A085000, A088217.

A180127 Upper bound for the determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

2, 32, 7414, 4993844, 5761178228, 11320943775475, 35966786849223443, 154715716383037989022, 1041732064414822689366009, 8436103376958505162325231670, 95816938885687281564299004113250, 1337411611273240103793149357629547975, 24089834168067078066162508828810807131186

Offset: 1

Hugo Pfoertner, Aug 12 2010

nonn

a(n) is an upper bound for A180128(n).

Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum JIPAM, Vol. 10, Iss. 3, Art. 63, 2008
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A180128 [Maximal determinant of matrix with first n^2 primes], A085000 [Maximal determinant of matrix with elements 1, ..., n^2], A180087 [Upper bound for A085000], A007504 [Sum of first n primes], A024450 [Sum of first n squares of primes].

a180127(n)={if(n<2,2, my(c=sum(k=1,n^2,prime(k))/n, d=sum(k=1,n^2,prime(k)^2)/n, t=(c^2-d)/(n-1)); floor(c*sqrt((d-t)^(n-1))))} \\ Hugo Pfoertner, Aug 27 2021

Let c = A007504(n^2)/n [(1/n)*sum of first n^2 primes]
and d = A024450(n^2)/n [(1/n)*sum of first n^2 squares of primes]
Then a(n) = floor(c*sqrt((d-t)^(n-1))) with t = (c^2-d)/(n-1).
log(a(n)) ~ (5*log(n) - log(3))*n/2 + n*log(log(n)). - Vaclav Kotesovec, Aug 28 2021

A301533 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; denominator.

Original entry on oeis.org

1, 12, 15120, 389188800, 64117105007155200, 10347762119105166852096000, 1389578338099539041754702978576000000, 4713072346356421489071058466945878500353772748800000000

Offset: 1

Hugo Pfoertner, Mar 23 2018

The maximum determinant achievable by arranging the fractions 1/1, 1/2, 1/3, ..., 1/n^2 as matrix entries is provided as fraction A301532(n) / a(n).

			See A301532.

Cf. A085000, A301371, A301532 (corresponding numerators).

A364203 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the integers from 1 to n^2.

Original entry on oeis.org

1, 0, 24, 0, 2736, 360144

Offset: 1

Stefano Spezia, Jul 13 2023

			The triangle begins:
  1;
  0,   24;
  0, 2736, 360144;
  ...

Cf. A085000 (maximal determinant), A088020 (row sums), A350565 (minimal permanent), A350566 (maximal permanent), A364206 (right diagonal).

Cf. A364226 (with prime numbers).

A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

Original entry on oeis.org

1, 24, 360144, 20914499571840

Offset: 1

Stefano Spezia, Jul 13 2023

Right diagonal of A364203.
Cf. A085000 (maximal determinant), A350565 (minimal permanent), A350566 (maximal permanent).
Cf. A364227 (with prime numbers).
Cf. A088020, A136609, A221976.

a(n) = (n^2)! - A221976(n). - Vaclav Kotesovec, Jul 16 2023

a(4) from Vaclav Kotesovec, Jul 16 2023 (using A221976)

A368539 Maximal sum of elements of A^2 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.

Original entry on oeis.org

1, 54, 761, 5284

Offset: 1

Sela Fried, Dec 29 2023

The next terms are at least (and probably equal to) 5284, 24303, 85352 and 248045.
The lower bounds for the terms a(4)-a(7) are confirmed. a(8) >= 626610, a(9) >= 1421271, a(10) >= 2959798, a(11) >= 5750977. - Hugo Pfoertner, Jan 21 2024
In addition to the conditions (a)-(d) described in para 2.2 of Fried and Mansour (2023), conjecturally optimal matrices found using simulated annealing have the following additional property: If, using simultaneous row and column rearrangement, the matrix is brought into a form in which the terms of the main diagonal are sorted in ascending order, then every single row and every single column is monotonically increasing. See the linked file for examples from n=2 to n=14. - Hugo Pfoertner, Jan 25 2024

			                                                     [1 3 4]
For n = 3, the sum of the elements of A^2, where A = [2 6 8], is 761.
                                                     [5 7 9]

Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
Hugo Pfoertner, Examples of solutions found by simulated annealing, for n=2-14. Jan 25, 2024.

Cf. A085000, A350566, A369396.

A180345 Lexicographically ordered 3X3 matrices containing numbers 1..9 with maximal determinant = 412.

Original entry on oeis.org

148726593, 157836492, 175429863, 184539762, 249715683, 267935481, 276418953, 294638751, 359814672, 368924571, 386517942, 395627841, 418953276, 429863175, 481267935, 492157836, 517942386, 539762184

Offset: 1

Zak Seidov, Jan 18 2011

The matrices are presented here as 9-digit decimal numbers, one digit per entry in the matrix.

There are exactly 36 such matrices: 148726593, 157836492, 175429863, 184539762, 249715683, 267935481, 276418953, 294638751, 359814672, 368924571, 386517942, 395627841, 418953276, 429863175, 481267935, 492157836, 517942386, 539762184, 571368924, 593148726, 627841395, 638751294, 672359814, 683249715, 715683249, 726593148, 751294638, 762184539, 814672359, 836492157, 841395627, 863175429, 924571368, 935481267, 942386517, 953276418.

			148726593 => {{1,4,8},{7,2,6},{5,9,3}}:
1 4 8
7 2 6
5 9 3
1*(2*3-9*6)-4(7*3-5*6)+8*(7*9-5*2)=412.

Cf. A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.

Cf. A088215 (1/36)*number of ways to express n as the determinant of a 3 X 3 matrix with elements 1..9.

A301370 Maximum determinant of an n X n (0,1)-matrix that has exactly 2*n ones.

Original entry on oeis.org

0, 2, 2, 3, 4, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64

Offset: 2

Hugo Pfoertner, Mar 20 2018

A proved upper bound is abs(a(n)) <= 6^(n/6), provided by Bruhn and Rautenbach. A conjectured sharper bound is abs(a(n)) <= 2^(n/3), provided by the same authors. For n=3*k, the bound is achieved by diagonally concatenating blocks ((1 1 0)(0 1 1)(1 0 1)).

The sharper bound is proved by Araujo, Balogh, and Wang in their article. See link. - Hugo Pfoertner, Nov 04 2020

			a(8) = 6 because no (0,1)-matrix with 2*8 ones with a greater determinant exists than
  ( 1 0 0 0 0 0 0 0 )
  ( 0 1 0 1 0 0 0 0 )
  ( 0 0 1 0 1 1 0 0 )
  ( 0 0 0 1 0 0 1 0 )
  ( 0 0 0 0 1 0 0 1 )
  ( 0 0 0 0 0 1 0 1 )
  ( 0 1 0 0 0 0 1 0 )
  ( 0 0 1 0 0 0 0 1 )

Igor Araujo, József Balogh, and Yuzhou Wang, Maximum determinant and permanent of sparse 0-1 matrices, arXiv:2011.01892 [math.CO], 3 Nov 2020.
Henning Bruhn and Dieter Rautenbach, Maximal determinants of combinatorial matrices, arXiv:1711.09935 [math.CO], 2017.
Mathoverflow, Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?, 2014-2018.
Yaroslav Shitov, On the determinant of a sparse 0-1 matrix, Linear Algebra and its Applications, Volume 554, 1 October 2018, Pages 49-50.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A003432, A017979, A085000.

A358485 a(n) is the maximal determinant of an n X n matrix using the integers 0 to n^2 - 1.

Original entry on oeis.org

1, 0, 6, 332, 36000, 6313388, 1765146660, 731664377274

Offset: 0

Stefano Spezia, Nov 18 2022

427402723914150 <= a(8) <= 427505414757161, 337815614862033534 <= a(9) <= 337888181610225000, 349880703121691699788 <= a(10) <= 349947469107433415221, with upper bounds from corollary 2 of Sigg (2018). - Hugo Pfoertner, Nov 21 2022

			a(3) = 332:
     [5, 7, 2;
      1, 3, 8;
      6, 0, 4]

Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 9 Apr 2018.

Cf. A085000 (integers 1 to n^2), A358486 (minimal permanent), A358487 (maximal permanent).

a(4)-a(6) from Hugo Pfoertner, Nov 19 2022

a(7) from Hugo Pfoertner, Nov 21 2022

cp's OEIS Frontend

A351148 a(n) is the maximal determinant of a symmetric n X n matrix using the integers 1 to n*(n+1)/2.

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A088745 Infinite array read by antidiagonals: for n>=1 let T_n = upper left n X n matrix. Then T_1 = (1), T_n has elements 1..n^2, contains T_{n-1} as n-1 X n-1 upper left submatrix and abs(det(T_n)) is a maximized.

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A180127 Upper bound for the determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

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A301533 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; denominator.

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A364203 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the integers from 1 to n^2.

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A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

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A368539 Maximal sum of elements of A^2 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.

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A180345 Lexicographically ordered 3X3 matrices containing numbers 1..9 with maximal determinant = 412.

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A301370 Maximum determinant of an n X n (0,1)-matrix that has exactly 2*n ones.

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A358485 a(n) is the maximal determinant of an n X n matrix using the integers 0 to n^2 - 1.

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