A309985 -id:A309985 - 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

A301371 Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.

Original entry on oeis.org

1, 1, 3, 18, 160, 2325, 41895, 961772, 27296640, 933251220

Offset: 0

Hugo Pfoertner, Mar 21 2018

929587995 <= a(9) <= 934173632 (upper bound from Gasper's determinant theorem). The lower bound corresponds to a Latin square provided in A309985, but it is unknown whether a larger determinant value can be achieved by an unconstrained arrangement of the matrix entries. - Hugo Pfoertner, Aug 27 2019

Oleg Vlasii found a 9 X 9 matrix significantly exceeding the determinant value achievable by a Latin square. See example and links. - Hugo Pfoertner, Nov 04 2020

			Matrices with maximum determinants:
a(2) = 3:
  (2  1)
  (1  2)
a(3) = 18:
  (3  1  2)
  (2  3  1)
  (1  2  3)
a(4) = 160:
  (4  3  2  1)
  (1  4  3  2)
  (3  1  4  3)
  (2  2  1  4)
a(5) = 2325:
  (5  3  1  2  4)
  (2  5  4  1  3)
  (4  1  5  3  2)
  (3  4  2  5  1)
  (1  2  3  4  5)
a(6) = 41895:
  (6  1  4  2  3  5)
  (3  6  2  1  5  4)
  (4  5  6  3  2  1)
  (5  3  1  6  4  2)
  (1  2  5  4  6  3)
  (2  4  3  5  1  6)
a(7) = 961772:
  (7  2  3  5  1  4  6)
  (3  7  6  4  2  1  5)
  (2  1  7  6  4  5  3)
  (4  5  1  7  6  3  2)
  (6  3  5  1  7  2  4)
  (5  6  4  2  3  7  1)
  (1  4  2  3  5  6  7)
a(8) = 27296640:
  (8  8  3  5  4  3  4  1)
  (1  8  6  3  1  6  6  5)
  (5  3  8  1  7  6  4  2)
  (5  1  6  8  2  4  7  3)
  (1  5  2  7  8  6  4  3)
  (7  3  2  4  3  8  2  7)
  (5  4  2  2  6  2  8  7)
  (4  5  7  6  5  1  1  7)
a(n) is an upper bound for the determinant of an n X n Latin square. a(n) = A309985(n) for n <= 7. a(8) > A309985(8). - _Hugo Pfoertner_, Aug 26 2019
From _Hugo Pfoertner_, Nov 04 2020: (Start)
a(9) = 933251220, achieved by a Non-Latin square:
  (9  5  5  3  3  2  2  8  8)
  (4  9  2  6  7  5  3  1  8)
  (4  7  9  2  1  8  6  3  5)
  (6  3  7  9  4  1  8  2  5)
  (6  2  8  5  9  7  1  4  3)
  (7  4  1  8  2  9  5  6  3)
  (7  6  3  1  8  4  9  5  2)
  (1  8  6  7  5  3  4  9  2)
  (1  1  4  4  6  6  7  7  9)
found by Oleg Vlasii as an answer to the IBM Ponder This Challenge November 2019. See links. (End)

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.
IBM Research, Large 9x9 determinant, Ponder This Challenge November 2019.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Oleg Vlasii, Determinant-OEIS-A301371-9, program and description, 4 Dec 2019.
Index entries for sequences related to maximal determinants

Cf. A085000, A309985, A328030, A328031.

A328030(n) <= a(n) <= A328031(n). - Hugo Pfoertner, Nov 04 2019

a(8) from Hugo Pfoertner, Aug 26 2019

a(9) from Hugo Pfoertner, Nov 04 2020

A308853 a(n) is the minimum absolute value of nonzero determinants of order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 126, 196, 144, 405

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jun 28 2019

We apply every symbol permutation on the representatives of isotopic classes to generate Latin squares of order n and calculate the determinants.

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

			For n=2, the only Latin squares of order 2 are [[1, 2], [2, 1]] and [[2, 1], [1, 2]].  Therefore, the minimum absolute value of the determinants of order 2 Latin squares is 3.

Brendan McKay, Latin squares
Hugo Pfoertner, Occurrence counts of determinant values for n=1..8, zipped (2019).
Eric Weisstein's World of Mathematics, Latin square
Wikipedia, Latin square
Index entries for sequences related to determinants

Cf. A040082, A301371 (upper bound for maximum determinant of Latin squares of order n), A309258, A309984, A309985.

# Takes a string and turns it into a square matrix of order n
def make_matrix(string,n):
    m = []
    row = []
    for i in range(0,n * n):
        if string[i] == '\n':
            continue
        if string[i] == ' ':
            continue
        row.append(Integer(string[i]) + 1)
        if len(row) == n:
            m.append(row)
            row = []
    return matrix(m)
# Reads a file and returns a list of the matrices in the file
def fetch_matrices(file_name,n):
    matrices = []
    with open(file_name) as f:
        L = f.readlines()
    for i in L:
        matrices.append(make_matrix(i,n))
    return matrices
# Takes a matrix and permutates each symbol in the matrix
# with the given permutation
def permute_matrix(matrix, permutation,n):
    copy = deepcopy(matrix)
    for i in range(0, n):
        for j in range(0 , n):
            copy[i,j] = permutation[copy[i][j] - 1]
    return copy
"""
Creates a determinant list with the following triples,
[Isotopy Class Representative, Permutation, Determinant]
The Isotopy class representatives come from a file that
contains all Isotopy classes.
"""
def create_determinant_list(file_name,n):
    the_list = []
    permu = (Permutations(n)).list()
    matrices = fetch_matrices(file_name,n)
    for i in range(0,len(matrices)):
        for j in permu:
            copy = permute_matrix(matrices[i],j,n)
            the_list.append([i,j,copy.determinant()])
            print(len(the_list))
    return the_list
# Froylan Maldonado, Jun 28 2019

a(8) from Hugo Pfoertner, Aug 24 2019

a(9) from Hugo Pfoertner, Aug 27 2019

A309258 a(n) is the number of distinct absolute values of determinants of order n Latin squares.

Original entry on oeis.org

1, 1, 1, 3, 6, 197, 3684, 159561

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 19 2019

We apply every symbol permutation on the representatives of isotopic classes to generate Latin squares of order n and calculated the determinants. We then obtained the absolute values of the determinants and removed duplicates.
These results are based on work supported by the National Science Foundation under grants numbered DMS-1852378 and DMS-1560019.
a(9) >= 1747706. - Hugo Pfoertner, Nov 20 2019

			For n = 5, the set of absolute values of determinants is {75, 825, 1200, 1575, 1875, 2325}, so a(5) = 6.

Froylan Maldonado, Code
Brendan McKay, Latin squares
Hugo Pfoertner, 8X8 Latin squares: Illustration of occurrence counts of determinant values, 5.7 MB, zoom in to see details (2019).
Hugo Pfoertner, Occurrence counts of determinant values for n=1..8, zipped (2019).

Cf. A040082, A088021, A301371, A308853, A309984, A309985.

Sage
```
# See Maldonado link.
```

a(8) from Hugo Pfoertner, Aug 26 2019

A328030 Maximum determinant of a circulant n X n matrix whose rows are permutations of [1,2,..,n].

Original entry on oeis.org

1, 1, 3, 18, 160, 2325, 41895, 961772, 26978400, 929587995, 36843728625, 1705290814194, 89802671542272, 5336424046419557, 354379732734283200, 26173529641406219400

Offset: 0

Hugo Pfoertner, Oct 02 2019

It is conjectured that A309985 is identical to this sequence. See Mathematics Stack Exchange link. The first rows of the corresponding circulant matrices are provided in A328029.

Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).
Wikipedia, Circulant matrix.

Cf. A309985, A328029, A328031.

a(n) = A309985(n) for n <= 9.

a(n) <= A328031(n).

a(15) from Hugo Pfoertner, Oct 14 2019

A328029 Lexicographically earliest permutation of [1,2,...,n] maximizing the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 2, 1, 4, 3, 1, 2, 4, 3, 5, 2, 1, 6, 3, 5, 4, 1, 2, 4, 6, 5, 3, 7, 2, 1, 5, 4, 8, 3, 6, 7, 1, 2, 4, 8, 6, 7, 5, 3, 9, 1, 2, 10, 7, 8, 3, 9, 5, 4, 6, 1, 2, 6, 11, 7, 9, 4, 8, 5, 3, 10, 2, 1, 7, 3, 12, 5, 9, 10, 4, 6, 11, 8, 1, 2, 12, 13, 5, 10, 6, 11, 3, 9, 8, 4, 7

Offset: 1

Hugo Pfoertner, Oct 02 2019

For n <= 9 the corresponding circulant matrices are n X n Latin squares with maximum determinant A309985(n). It is conjectured that this also holds for n > 9. See Mathematics Stack Exchange link.

			The triangle starts
  1;
  2,  1;
  1,  2,  3;
  2,  1,  4,  3;
  1,  2,  4,  3,  5;
  2,  1,  6,  3,  5,  4;
  1,  2,  4,  6,  5,  3,  7;
  2,  1,  5,  4,  8,  3,  6,  7;
  1,  2,  4,  8,  6,  7,  5,  3,  9;
  1,  2, 10,  7,  8,  3,  9,  5,  4,  6;
.
The 4th row of the triangle T(4,1)..T(4,4) = a(7)..a(10) is [2,1,4,3] because this is the lexicographically earliest permutation of [1,2,3,4] producing a circulant 4 X 4 matrix with maximum determinant A328030(4) = 160.
  [2, 1, 4, 3;
   3, 2, 1, 4;
   4, 3, 2, 1;
   1, 4, 3, 2].
All lexicographically earlier permutations lead to smaller determinants, with [1,2,3,4] and [1,4,3,2] producing determinants = -160.

Hugo Pfoertner, Table of n, a(n) for n = 1..120, rows 1..15 of triangle, flattened
Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).
Wikipedia, Circulant matrix.

Cf. A301371, A309985, A328030, A328031, A328062.

f[n_] := (p = Permutations[Table[i, {i, n}]]; L = Length[p]; det = Max[Table[Det[Reverse /@ Partition[p[[i]], n, 1, {1, 1}]], {i, 1, L}]]; mat = Table[Reverse /@ Partition[p[[i]], n, 1, {1, 1}], {i, 1, L}]);
n = 1; While[n <= 10, ClearSystemCache[[]]; f[n]; triangle = Parallelize[Select[mat, Max[Det[#]] == det &]]; Print[SortBy[triangle, Less][[1]][[1]]]; n++]; (* Kebbaj Mohamed Reda, Dec 03 2019; edited by Michel Marcus, Dec 24 2023 *)

A309984 Number of n X n Latin squares with determinant 0, divided by 2.

Original entry on oeis.org

0, 0, 0, 16, 0, 2088, 5752, 199600889

Offset: 1

Hugo Pfoertner, Aug 26 2019

			a(4)=16: There are 2*a(4) = 32 4 X 4 Latin squares with determinant = 0, one of which is
  [1  4  3  2]
  [4  1  2  3]
  [3  2  1  4]
  [2  3  4  1].
An example of a 6 X 6 Latin square with determinant = 0 is
  [1  3  4  6  5  2]
  [3  2  6  5  4  1]
  [4  6  3  2  1  5]
  [6  5  1  3  2  4]
  [5  4  2  1  3  6]
  [2  1  5  4  6  3].

Brendan McKay, Latin squares.
Hugo Pfoertner, Occurrence counts of determinant values of Latin squares for n=1..8, zipped (2019).

Cf. A040082, A136609, A221976, A301371, A308853, A309258, A309985.

A328031 Upper bound for the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,...,n^n}.

Original entry on oeis.org

1, 1, 3, 18, 172, 2343, 42439, 976050, 27583338, 934173632, 37180409223, 1711870023666, 90007747560742, 5346164992890599, 355442084718552178, 26244000000000000000, 2137205155719002036203, 190811368062993357765186, 18577775646585813239195436, 1963166636163973976912956096

Offset: 0

Hugo Pfoertner, Oct 02 2019

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

Cf. A301371, A309985, A328030.

for(n=1,20,print1(floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)),", "))

a(n) = floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)) (Corollary 3 in M. Sigg's article).

cp's OEIS Frontend

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

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A301371 Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.

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A308853 a(n) is the minimum absolute value of nonzero determinants of order n Latin squares.

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A309258 a(n) is the number of distinct absolute values of determinants of order n Latin squares.

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A328030 Maximum determinant of a circulant n X n matrix whose rows are permutations of [1,2,..,n].

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A328029 Lexicographically earliest permutation of [1,2,...,n] maximizing the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

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A309984 Number of n X n Latin squares with determinant 0, divided by 2.

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A328031 Upper bound for the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,...,n^n}.

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