A309258 -id:A309258 - cp's OEIS frontend

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

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

A309985 Maximum determinant of an n X n Latin square.

Original entry on oeis.org

1, 1, 3, 18, 160, 2325, 41895, 961772, 26978400, 929587995

Offset: 0

Hugo Pfoertner, Aug 26 2019

a(n) = A301371(n) for n <= 7. a(8) < A301371(8) = 27296640, a(9) < A301371(9) = 933251220.
a(10) = 36843728625, conjectured. See Stack Exchange link. - Hugo Pfoertner, Sep 29 2019
A328030(n) <= a(n) <= A301371(n). - Hugo Pfoertner, Dec 02 2019
It is unknown, but very likely, that A301371(n) > a(n) also holds for all n > 9 - Hugo Pfoertner, Dec 12 2020

			An example of an 8 X 8 Latin square with maximum determinant is
  [7  1  3  4  8  2  5  6]
  [1  7  4  3  6  5  2  8]
  [3  4  1  7  2  6  8  5]
  [4  3  7  1  5  8  6  2]
  [8  6  2  5  4  7  1  3]
  [2  5  6  8  7  3  4  1]
  [5  2  8  6  1  4  3  7]
  [6  8  5  2  3  1  7  4].
An example of a 9 X 9 Latin square with maximum determinant is
  [9  4  3  8  1  5  2  6  7]
  [3  9  8  5  4  6  1  7  2]
  [4  1  9  3  2  8  7  5  6]
  [1  2  4  9  7  3  6  8  5]
  [8  3  5  6  9  7  4  2  1]
  [2  7  1  4  6  9  5  3  8]
  [5  8  6  7  3  2  9  1  4]
  [7  6  2  1  5  4  8  9  3]
  [6  5  7  2  8  1  3  4  9].
An example of a 10 X 10 Latin square with abs(determinant) = 36843728625 is a circulant matrix with first row [1, 3, 7, 9, 8, 6, 5, 4, 2, 10], but it is not known if this is the best possible. - _Kebbaj Mohamed Reda_, Nov 27 2019 (reworded by _Hugo Pfoertner_)

Brendan McKay, Latin squares.
Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).

Cf. A040082, A301371, A308853, A309258, A309984, A328029, A328030.

a(9) from Hugo Pfoertner, Aug 30 2019

a(0)=1 prepended by Alois P. Heinz, Oct 02 2019

A322576 Least nonnegative integer that cannot be expressed as 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

0, 1, 9, 139, 2111, 40021, 942937, 27003797

Offset: 1

Hugo Pfoertner, Aug 29 2019

			a(1) = 0 because det[1] = 1.
a(2) = 1 because det[1,1; 2,2] = 0 and det[2,1; 1,2] = 3 are the only determinant values >= 0 that can be made by permuting the matrix entries {1,1, 2,2}.
a(3) = 9, because it is the first missing value in the list of A309799(3) = 13 determinant values corresponding to {1,1,1, 2,2,2, 3,3,3}: 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 15, 18.

Cf. A088216, A301371, A309258, A309799, A327281.

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.

A309259 a(n) is the greatest common divisor of the determinants of order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 63, 196, 144, 405

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 calculate the determinants. We then compute the greatest common divisor of the values obtained.

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

			For n=4, the set of absolute values of the determinants is {0, 80, 160}, so the greatest common divisor of the determinants is 80. Therefore, a(4)=80.

Peterson Lenard, Greatest Common Divisor of all determinants
Brendan McKay, Combinatorial Data

Cf. A301371, A308853, A309088, A309258.

Sage
```
# See Peterson Lenard link
```

a(8), a(9) from Hugo Pfoertner, Sep 02 2019

A309799 Number of distinct nonnegative values that can be assumed by 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, 2, 13, 147, 2162, 40498, 948618

Offset: 1

Hugo Pfoertner, Aug 29 2019

a(8) >= 27091220. - Hugo Pfoertner, Sep 23 2019

			a(2) = 2: 0 = det[1,1; 2,2], 3 = det[2,1; 1,2] are the two possible nonnegative values of the determinant.
a(3) = 13, because
   0 = det[1,2,3; 1,2,3; 1,2,3],  1 = det[2,2,1; 3,2,1; 3,3,1],
   2 = det[3,2,3; 1,2,3; 1,1,2],  3 = det[3,3,3; 1,2,2; 1,1,2],
   4 = det[1,3,3; 2,2,1; 1,3,2],  5 = det[2,2,1; 1,3,3; 1,2,3],
   6 = det[1,3,2; 1,2,3; 2,1,3],  7 = det[1,3,1; 1,2,3; 2,2,3],
   8 = det[1,1,2; 3,3,2; 1,3,2], 12 = det[2,3,1; 2,1,3; 3,1,2],
  13 = det[3,3,1; 1,3,2; 2,1,2], 15 = det[2,1,3; 3,1,1; 2,3,2],
  18 = det[2,3,1; 1,2,3; 3,1,2]
are the 13 possible nonnegative values of the determinant.

Cf. A088216, A088217, A301371, A309258, A322576, A327281.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions

A309985 Maximum determinant of an n X n Latin square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A322576 Least nonnegative integer that cannot be expressed as the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n, ..., n^n}.

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

A309259 a(n) is the greatest common divisor of the determinants of order n Latin squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions

A309799 Number of distinct nonnegative values that can be assumed by the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,..,n^n}.

Views

Author

Keywords

Comments

Examples

Crossrefs