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A308853 a(n) is the minimum absolute value of nonzero determinants of order n Latin squares. 8
1, 3, 18, 80, 75, 126, 196, 144, 405 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..9.

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


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.



# 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':


        if string[i] == ' ':


        row.append(Integer(string[i]) + 1)

        if len(row) == n:


            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()])


    return the_list

# Froylan Maldonado, Jun 28 2019


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

Sequence in context: A056319 A056310 A309259 * A309257 A135371 A086346

Adjacent sequences:  A308850 A308851 A308852 * A308854 A308855 A308856




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


a(8) from Hugo Pfoertner, Aug 24 2019

a(9) from Hugo Pfoertner, Aug 27 2019



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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)