A309984 -id:A309984 - 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

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

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

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A309985 Maximum determinant of an n X n Latin square.

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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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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions