A350566 -id:A350566 - cp's OEIS frontend

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

1, 10, 412, 40800, 6839492, 1865999570, 762150368499

Offset: 1

Robert G. Wilson v, Jun 16 2003

Bounds for the next terms and the corresponding matrices are given by O. Gasper, H. Pfoertner and M. Sigg: 440960274696935 <= a(8) < 441077015225642, 346254605664223620 <= a(9) < 346335386150480625, 356944784622927045792 <= a(10) < 357017114947987625629. a(n) < sqrt(3*((n^5+n^4+n^3+n^2)/12)^n*(n^2+1)/(n+1)). - Hugo Pfoertner, Aug 15 2010
Improved lower bounds (private communication from Benjamin R. Buhrow, Dec 09 2019): a(8) >= 440970981670289, a(9) >= 346260899916111296. - Hugo Pfoertner, Jan 25 2021
Improved lower bound (private communication from Richard Gosiorovsky, Aug 18 2021): a(10) >= 356948996371054862392. - Hugo Pfoertner, Aug 24 2021

			The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is -412):
   9 3 5
   4 8 1
   2 6 7
Results from a specially adapted hill-climbing algorithm strongly suggest that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, a(n) is approximately 0.44*n^(2.06*n), suggesting that a(7) is close to 6.8*10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 2003
a(5) confirmed by _Robert Israel_ and _Hugo Pfoertner_. A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 20 5) (10 13 2 22 19) (16 1 18 8 21) ). - _Hugo Pfoertner_, Sep 23 2003
a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ). - _Hugo Pfoertner_, Sep 23 2003
a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.

Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.
Hugo Pfoertner, Maximal determinant of matrix with elements 1..n. FORTRAN program.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Square Matrix
Index entries for sequences related to maximal determinants

Cf. A088214, A088215, A088216, A088217, A088237, A180087 [upper bounds for a(n)], A180128, A221976, A301371, A301532, A301533, A309985, A325900, A350566.

Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], {k, n2!}]; {dMax, mMax} (* T. D. Noe *)

vectomat(v)=my(n=sqrtint(#v));matrix(n,n,i,j,v[n*(i-1)+j])
a(n)=my(m,t,M); n*=n; for(k=0,(n-1)!-1, t=matdet(M=vectomat(numtoperm(n,k))); if(abs(t)>m, m=abs(t); print(t" "M)));m \\ Charles R Greathouse IV, Sep 13 2013

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
a(6) from Hugo Pfoertner, Sep 23 2003
Entry edited by N. J. A. Sloane, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.
a(7) from Hugo Pfoertner, Jan 22 2008

A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

Original entry on oeis.org

1, 1, 10, 436, 52197, 13300936, 6192060119

Offset: 0

Stefano Spezia and Hugo Pfoertner, Jan 20 2022

a(7) <= 4755379618016 from the matrix
  [ 1,  2,  3,  4,  5,  6,  7;
    8, 14, 19, 23, 29, 33, 36;
    9, 15, 21, 26, 32, 37, 41;
   10, 16, 22, 28, 35, 40, 44;
   11, 17, 24, 30, 38, 43, 46;
   12, 18, 25, 31, 39, 45, 48;
   13, 20, 27, 34, 42, 47, 49]. - Pontus von Brömssen, Aug 30 2025

			a(2) = 10:
  [1, 3;
   2, 4]
.
a(3) = 436:
  [1, 3, 2;
   4, 8, 6;
   5, 9, 7]
.
a(4) = 52197:
  [1,  2,  4,  3;
   6,  9, 15, 12;
   5,  8, 13, 11;
   7, 10, 16, 14]
.
a(5) = 13300936:
  [16,  8, 24, 21, 12;
   18,  9, 25, 23, 13;
    3,  1,  5,  4,  2;
   14,  6, 20, 17, 10;
   15,  7, 22, 19, 11]
.
a(6) = 6192060119:
  [36, 35, 33, 31, 27,  6;
   11, 10,  9,  8,  7,  1;
   34, 32, 30, 28, 25,  5;
   22, 21, 19, 18, 16,  3;
   29, 26, 24, 23, 20,  4;
   17, 15, 14, 13, 12,  2]

Carl-Erik Fröberg, On a combinatorial problem related to permanents, BIT 28 (1988), No. 3, 406-411.

Cf. A085000, A350566 (maximum), A350858, A350859, A358486 (elements 0 to n^2-1).

from itertools import permutations
from sympy import Matrix
def A350565(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

A350859 Maximal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

1, 2, 41, 11868, 12124850, 25767879812, 101120963518528

Offset: 0

Stefano Spezia, Jan 19 2022

			a(2) = 41:
   5    2
   3    7
a(3) = 11868:
  23    5    3
   2   13   19
   7   17   11

Cf. A114533, A180128, A350566, A350858 (minimal).

from itertools import permutations
from sympy import Matrix
def A350859(n): return 1 if n == 0 else max(Matrix(n,n,p).per() for p in permutations(prime(m) for m in range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

a(4)-a(6) from Hugo Pfoertner, Jan 21 2022

A358487 a(n) is the maximal permanent of an n X n matrix using the integers 0 to n^2 - 1.

Original entry on oeis.org

1, 0, 6, 553, 107140, 40179728, 27312009708

Offset: 0

Stefano Spezia, Nov 18 2022

			a(3) = 553:
     [1, 2, 8;
      7, 5, 0;
      4, 6, 3]

Cf. A350566 (integers 1 to n^2), A358486 (minimal).

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

A351611 Maximal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 11, 420, 41451, 7985639, 2779152652

Offset: 0

Stefano Spezia, Feb 14 2022

			a(3) = 420:
    1    5    6
    5    3    4
    6    4    2
a(4) = 41451:
    1    5    8   10
    5    4    9    7
    8    9    3    6
   10    7    6    2

Cf. A000217, A351153, A351610 (minimal).

Cf. A350566, A350859, A350938, A350940, A351020, A351022.

a(5)-a(6) from Hugo Pfoertner, Feb 15 2022

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.

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

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A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

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A350859 Maximal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

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A358487 a(n) is the maximal permanent of an n X n matrix using the integers 0 to n^2 - 1.

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A351611 Maximal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

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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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