A040082 -id:A040082 - cp's OEIS frontend

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

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
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# See Maldonado link.
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a(8) from Hugo Pfoertner, Aug 26 2019

A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

Original entry on oeis.org

1, 1, 4, 456, 10936320, 130025295912960, 2209617218725251404267520

Offset: 0

Joshua Zucker and Joe Keane

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(2*n). - Max Alekseyev, Apr 23 2010

a(5)-a(6) from Ian Wanless, Oct 20 2019

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.

A000528 Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 17, 324, 842227, 57810418543, 104452188344901572, 6108088657705958932053657

Offset: 1

N. J. A. Sloane

Here "type" means an equivalence class of Latin squares under the operations of row permutation, column permutation, symbol permutation and transpose. In the 1-factorizations formulation, these operations are labeling of left side, labeling of right side, permuting the order in which the factors are listed and swapping the left and right sides, respectively. - Brendan McKay

There are 6108088657705958932053657 isomorphism classes of one-factorizations of K_{11,11}. - Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009

CRC Handbook of Combinatorial Designs, 1996, p. 660.
Denes and Keedwell, Latin Squares and Applications, Academic Press 1974.

A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219
B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs, to appear (2005).
Index entries for sequences related to Latin squares and rectangles

See A040082 for another version.

Cf. A002860, A003090, A000315, A040082, A000479.

More terms from Richard Bean, Feb 17 2004

a(11) from Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009

A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

Original entry on oeis.org

1, 1, 2, 6, 96, 720, 328320, 31449600, 440952422400, 444733651353600, 471835793808949248000, 10070314878246926155776000, 1058410183156945383046388908032000, 614972203951464612786852376432607232000

Offset: 0

Joshua Zucker and Joe Keane

The even and odd subsequences are A036980, A036981.

			a(3) = 6 because the first row is arbitrary (say, 213) and the rest is then determined. By symmetry the second row has to be 132 or 123 but in order for the third row/column to work it has to be 132.

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(n) * n!. [From Max Alekseyev, Apr 23 2010]

a(10)-a(13) (using A035481) from Alois P. Heinz, May 05 2023

A162544 Triangle giving number L(k,n) of isotopy classes of Latin rectangles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 4, 16, 56, 40, 22, 1, 4, 56, 1398, 6941, 3479, 564, 1, 7, 370, 93561, 4735238, 29163047, 13302311, 1676267, 1, 8, 2877, 8024046

Offset: 1

Vladan Radivojevic (vladanradivojevic(AT)yahoo.com), Jul 05 2009

Isotopy class means an equivalence class of Latin rectangles under the operations of row permutation, column permutation and symbol permutation.

B. D. Mckay, Isomorph-free exhaustive generation, Department of Computer Science Australian National University, Canberra, 1998.

Cf. A001009, A040082.

A264603 Number of structurally distinct Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 12, 192, 145164, 1524901344

Offset: 1

N. J. A. Sloane, Nov 23 2015

"Structurally distinct" means that the squares cannot be made identical by means of rotation, reflection, and/or permutation of the symbols. For other notions of distinctness, see A000479 and A040082.

Hendrik Willem Barink, Email to N. J. A. Sloane, Nov 22 2015

Hendrik Willem Barink, How many structurally different latin squares do exist?
Gabriel Crudele, Peter Dukes, and Jonathan A. Noel, Six Permutation Patterns Force Quasirandomness, arXiv:2303.04776 [math.CO], 2023.
Mathematics Stack Exchange, How many structurally different latin squares of order 5 exist?
Mathematics Stack Exchange, How many structurally distinct Latin squares are there of order 7?
Index entries for sequences related to Latin squares and rectangles

Cf. A000479, A040082.

a(7) from MSE link, added by Max Alekseyev, Sep 01 2017

A162545 Triangle giving number L(k,n) of normalized Latin rectangles which are lexicographically minimum of its isotopy class.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 4, 14, 34, 31, 22, 1, 4, 54, 427, 1410, 1096, 564, 1, 7, 330, 20259, 509027, 3144797, 2847673, 1676267, 1, 8, 2409

Offset: 1

Vladan Radivojevic (vladanradivojevic(AT)yahoo.com), Jul 05 2009

Cf. A001009, A040082, A162544.

cp's OEIS Frontend

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

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A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

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

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A000528 Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.

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A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

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A162544 Triangle giving number L(k,n) of isotopy classes of Latin rectangles.

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A264603 Number of structurally distinct Latin squares of order n.

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A162545 Triangle giving number L(k,n) of normalized Latin rectangles which are lexicographically minimum of its isotopy class.

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Author

Keywords

Crossrefs