A287650 -id:A287650 - cp's OEIS frontend

A287649 Number of horizontally symmetric diagonal Latin squares of order 2n with the first row in ascending order.

0, 2, 64, 3612672, 82731715264512

Offset: 1

Eduard I. Vatutin, May 29 2017

The number of horizontally symmetric diagonal Latin squares (X) is equal to the number of vertically symmetric diagonal Latin squares. The total number of diagonal Latin squares with either horizontal or vertical symmetry (see A296060) is equal to 2*X-Y, where Y is the number of doubly symmetric diagonal Latin squares (see A287650). - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

The sum of symmetric elements a[i, j] and a[i, n-1-j] in a horizontally symmetric normalized square of order n is constant and equal to n-1 for all pairs of elements (with rows and columns numbered from 0 to n-1 and elements values from 0 to n-1). This is not true for vertically symmetric normalized squares. - Eduard I. Vatutin, Oct 19 2017

			Horizontally symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 0 5 3 1
  5 4 3 2 1 0
  2 5 4 1 0 3
  3 0 1 4 5 2
  1 3 5 0 2 4
Vertically symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 5 0 3 1
  3 5 1 2 0 4
  5 3 0 4 1 2
  2 4 3 1 5 0
  1 0 4 5 2 3

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, continuation (in Russian)
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian)
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 30-36 (in Russian)
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A003191, A287650, A292516, A296060, A296061, A340546.

a(n) = A292516(n) / (2*n)!.

a(n) = (A296060(n) + A287650(n/2))/2 for even n; a(n) = A296060(n)/2 for odd n. - Andrew Howroyd, May 28 2021

a(5) calculated and added by Eduard I. Vatutin, S. E. Kochemazov and O. S. Zaikin, Jun 15 2017

A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order.

Original entry on oeis.org

1, 0, 0, 2, 8, 0, 2816, 135168, 327254016

Offset: 1

Eduard I. Vatutin, Oct 16 2017

A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i][j] and a[n-1-i][n-1-j] (with rows and columns numbered from 0 to n-1).
a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist). It seems that a(n)=0 for n == 2 (mod 4).
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(4n) >= A287650(n). - Eduard I. Vatutin, May 03 2021

			0 1 2 3 4 5 6 7 8
6 3 0 2 7 8 1 4 5
3 2 1 8 6 7 0 5 4
7 8 6 5 1 3 4 0 2
8 6 4 7 2 0 5 3 1
2 7 5 6 8 4 3 1 0
5 4 7 0 3 1 8 2 6
4 5 8 1 0 2 7 6 3
1 0 3 4 5 6 2 8 7

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles

Cf. A287649, A287650, A293778, A340545.

a(n) = A293778(n) / n!.

A292516 Number of horizontally symmetric diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 48, 46080, 145662935040, 300216848351861145600

Offset: 1

Eduard I. Vatutin, Sep 18 2017

The number of horizontally symmetric diagonal Latin squares (X) is equal to the number of vertically symmetric diagonal Latin squares. The total number of symmetric diagonal Latin squares is equal to 2*X-Y, where Y is a number of double symmetric diagonal Latin squares (sequence A292517). - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

			Horizontally symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 0 5 3 1
  5 4 3 2 1 0
  2 5 4 1 0 3
  3 0 1 4 5 2
  1 3 5 0 2 4
Vertically symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 5 0 3 1
  3 5 1 2 0 4
  5 3 0 4 1 2
  2 4 3 1 5 0
  1 0 4 5 2 3

E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19 (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 30-36 (in Russian)
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A003191, A287650, A287649, A296060, A296061, A340546.

a(n) = A287649(n) * (2*n)!.

A292517 Number of doubly symmetric diagonal Latin squares of order 4n.

Original entry on oeis.org

48, 495452160, 38903149816763645952000, 127654439655255918929515331054014121902080000

Offset: 1

Eduard I. Vatutin, Sep 18 2017

A doubly symmetric square has symmetries in both horizontal and vertical planes.
The plane symmetry requires one-to-one correspondence between the values of elements a[i,j] and a[N+1-i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1-j] in a horizontal plane for 1 <= i,j <= N. - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A293778(4n). - Eduard I. Vatutin, May 03 2021

			Doubly symmetric diagonal Latin square example:
0 1 2 3 4 5 6 7
3 2 7 6 1 0 5 4
2 3 1 0 7 6 4 5
6 7 5 4 3 2 0 1
7 6 3 2 5 4 1 0
4 5 0 1 6 7 2 3
5 4 6 7 0 1 3 2
1 0 4 5 2 3 7 6
In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.
In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.

A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 30-36. (in Russian)
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles

Cf. A003191, A287649, A287650, A293778, A340550.

a(n) = A287650(n) * (4n)!.

a(2) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Edited and a(3) from A287650 added by Max Alekseyev, Aug 23 2018, Sep 07 2018
a(4) from Andrew Howroyd, May 31 2021

A296060 Number of one-plane symmetric diagonal Latin squares of order 2n with first row 0,1,...,2n-1.

Original entry on oeis.org

0, 2, 128, 7213056

Offset: 1

Eduard I. Vatutin, Dec 04 2017

One-plane symmetric diagonal Latin squares are vertically or horizontally symmetric diagonal Latin squares. a(n) is equal to 2*X-Y, where X is the number of horizontally symmetric diagonal Latin squares with constant first row (sequence A287649), and Y is the number of doubly symmetric diagonal Latin squares with constant first row (sequence A287650).

			A horizontally symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 0 5 3 1
  5 4 3 2 1 0
  2 5 4 1 0 3
  3 0 1 4 5 2
  1 3 5 0 2 4
A vertically symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 5 0 3 1
  3 5 1 2 0 4
  5 3 0 4 1 2
  2 4 3 1 5 0
  1 0 4 5 2 3
A doubly symmetric diagonal Latin square:
  0 1 2 3 4 5 6 7
  3 2 7 6 1 0 5 4
  2 3 1 0 7 6 4 5
  6 7 5 4 3 2 0 1
  7 6 3 2 5 4 1 0
  4 5 0 1 6 7 2 3
  5 4 6 7 0 1 3 2
  1 0 4 5 2 3 7 6

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Corrections, Intellectual and Information Systems (2017), pp. 30-36 (in Russian).
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles

Cf. A287649, A287650, A292516, A296061, A340546.

a(n) = 2*A287649(n) - A287650(n).

A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 47, 0, 0, 0

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A doubly symmetric square has symmetries in both the horizontal and vertical planes (see A292517).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A340545(n). - Eduard I. Vatutin, May 28 2021

			An example of a doubly symmetric diagonal Latin square:
  0 1 2 3 4 5 6 7
  3 2 7 6 1 0 5 4
  2 3 1 0 7 6 4 5
  6 7 5 4 3 2 0 1
  7 6 3 2 5 4 1 0
  4 5 0 1 6 7 2 3
  5 4 6 7 0 1 3 2
  1 0 4 5 2 3 7 6
In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.
In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A292517, A287650, A340545, A340546.

Name clarified by Andrew Howroyd, Oct 22 2023

A357473 Number of types of generalized symmetries in diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 10, 8, 12, 12

Offset: 1

Eduard I. Vatutin, Sep 29 2022

The diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv — some permutations that are describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). Diagonal Latin squares with generalized symmetries are rare; usually they have a large number of transversals, orthogonal mates, etc.
a(n) <= A000041(n)^3. - Eduard I. Vatutin, Dec 29 2022
For all orders in which diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 22 2023
The set of the generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so a(n) <= A358515(n). - Eduard I. Vatutin, Jan 24 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so A358394(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023

			For order n=5 there are 7 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1,1},
  2 - {1,1,1,2},
  3 - {1,1,3},
  4 - {1,2,2},
  5 - {1,4},
  6 - {2,3},
  7 - {5}.
Diagonal Latin squares of order n=5 has a(5)=8 different types of generalized symmetries:
1. A=0123442301341201304220413 (string representation of the square), Px=[0,1,2,3,4], Py=[0,1,2,3,4], Pv=[0,1,2,3,4] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type (1,1,1).
2. A=0123442301341201304220413, Px=[0,1,2,3,4], Py=[1,3,0,4,2], Pv=[1,3,0,4,2], L(Px)={1,1,1,1,1}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (1,7,7).
3. A=0123442013143203014223401, Px=[0,3,2,4,1], Py=[1,4,2,3,0], Pv=[1,4,2,3,0], L(Px)={1,1,3}, L(Py)={1,1,3}, L(Pv)={1,1,3}, generalized symmetry type (3,3,3).
4. A=0123442301341201304220413, Px=[0,2,1,4,3], Py=[0,2,1,4,3], Pv=[0,2,1,4,3], L(Px)={1,2,2}, L(Py)={1,2,2}, L(Pv)={1,2,2}, generalized symmetry type (4,4,4).
5. A=0123442301341201304220413, Px=[0,3,4,2,1], Py=[0,3,4,2,1], Pv=[0,3,4,2,1], L(Px)={1,4}, L(Py)={1,4}, L(Pv)={1,4}, generalized symmetry type (5,5,5).
6. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[0,1,2,3,4], Pv=[4,2,3,0,1], L(Px)={5}, L(Py)={1,1,1,1,1}, L(Pv)={5}, generalized symmetry type (7,1,7).
7. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[3,4,1,2,0], Pv=[0,1,2,3,4], L(Px)={5}, L(Py)={5}, L(Pv)={1,1,1,1,1}, generalized symmetry type (7,7,1).
8. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[1,3,0,4,2], Pv=[2,0,4,1,3], L(Px)={5}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (7,7,7).

Eduard I. Vatutin, About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7, 28 Jul 2022.
Eduard I. Vatutin, Proving lists (4, 5, 6, 7), Jul 28 2022.
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A358515, A358394, A358891.

A358394 Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 10, 7, 0, 8

Offset: 1

Eduard I. Vatutin, Nov 20 2022

An orthogonal diagonal Latin square is a square that has at least one orthogonal diagonal mate.
A diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv are some permutations that describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). Diagonal Latin squares with generalized symmetries are rare; usually they have a large number of transversals, orthogonal mates, etc.
a(8) >= 74, a(9) >= 41, a(10) >= 27.
a(n) <= A000041(n)^3. - Eduard I. Vatutin, Jan 01 2023
For all orders in which orthogonal diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 22 2023
The set of the generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so a(n) <= A358891(n). - Eduard I. Vatutin, Jan 24 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so a(n) <= A357473(n). - Eduard I. Vatutin, Jan 25 2023

			For order n=5 there are 7 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1,1},
  2 - {1,1,1,2},
  3 - {1,1,3},
  4 - {1,2,2},
  5 - {1,4},
  6 - {2,3},
  7 - {5}.
The diagonal Latin square
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
of order n=5 has all a(5)=7 possible different types of generalized symmetries:
1. Px=[0,1,2,3,4], Py=[0,1,2,3,4], Pv=[0,1,2,3,4] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type (1,1,1).
2. Px=[0,1,2,3,4], Py=[1,2,3,4,0], Pv=[1,2,3,4,0], L(Px)={1,1,1,1,1}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (1,7,7).
3. Px=[0,4,3,2,1], Py=[0,4,3,2,1], Pv=[0,4,3,2,1], L(Px)={1,2,2}, L(Py)={1,2,2}, L(Pv)={1,2,2}, generalized symmetry type (4,4,4).
4. Px=[0,2,4,1,3], Py=[0,2,4,1,3], Pv=[0,2,4,1,3], L(Px)={1,4}, L(Py)={1,4}, L(Pv)={1,4}, generalized symmetry type (5,5,5).
5. Px=[1,2,3,4,0], Py=[0,1,2,3,4], Pv=[2,3,4,0,1], L(Px)={5}, L(Py)={1,1,1,1,1}, L(Pv)={5}, generalized symmetry type (7,1,7).
6. Px=[1,2,3,4,0], Py=[3,4,0,1,2], Pv=[0,1,2,3,4], L(Px)={5}, L(Py)={5}, L(Pv)={1,1,1,1,1}, generalized symmetry type (7,7,1).
7. Px=[1,2,3,4,0], Py=[1,2,3,4,0], Pv=[3,4,0,1,2], L(Px)={5}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (7,7,7).

Eduard I. Vatutin, About the number of types of generalized symmetries in orthogonal diagonal Latin squares of orders 1-5.
Eduard I. Vatutin, About the number of types of generalized symmetries in orthogonal diagonal Latin squares of orders 7-9.
Eduard I. Vatutin, About the number of types of generalized symmetries in orthogonal diagonal Latin squares of orders 10.
Eduard I. Vatutin, Proving lists (4, 5, 7, 8, 9, 10), Jul 31 2022
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A358515, A357473, A358891.

A358515 Number of types of generalized symmetries in diagonal Latin squares of order n in parastrophic slices.

Original entry on oeis.org

6, 0, 0, 76, 74, 199, 861

Offset: 1

Eduard I. Vatutin, Nov 20 2022

A diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv are some permutations that describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). A diagonal Latin square A has a generalized symmetry in parastrophic slices if for all cells A[x][y] = v and A[a'][b'] = c' the relation is satisfied: (a',b',c') = R(Px[x],Py[y],Pv[v]), where R is one of 6 possible parastrophic transformations:
   1. (x,y,v) -> (a,b,c) (trivial).
   2. (x,v,y) -> (a,b,c).
   3. (y,x,v) -> (a,b,c) (transpose).
   4. (y,v,x) -> (a,b,c).
   5. (v,x,y) -> (a,b,c).
   6. (v,y,x) -> (a,b,c).
A set of squares with selected parastrophic transformation R forms one of 6 parastrophic slices. Diagonal Latin squares with a generalized symmetry are a special case of generalized symmetries in parastrophic slice # 1. Diagonal Latin squares with generalized symmetries in parastrophic slices are rare; usually they have a large number of transversals, orthogonal mates, etc.
For all orders in which diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) in first parastrophic slice and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 24 2023, updated Mar 25 2023
The set of generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so A357473(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so A358891(n) <= a(n). - Eduard I. Vatutin, Jan 28 2023

			For order n=4 there are 5 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1},
  2 - {1,1,2},
  3 - {1,3},
  4 - {2,2},
  5 - {4}.
Diagonal Latin squares of order n=4 have a(4)=76 different types of generalized symmetries in parastrophic slices.
Slice 1 (10 generalized symmetries), R=(x,y,v):
  1. A=0123321010322301 (string representation of the square), Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 1-(1,1,1).
  2. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 1-(1,4,4).
  ...
  10. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 1-(5,5,2).
Slice 2 (10 generalized symmetries), R=(x,v,y):
  11. A=0123321010322301, Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3], L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 2-(1,1,1).
  12. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 2-(1,4,4).
  ...
  20. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 2-(5,5,2).
Slice 3 (14 generalized symmetries).
Slice 4 (14 generalized symmetries).
Slice 5 (14 generalized symmetries).
Slice 6 (14 generalized symmetries).
Total 10+10+14+14+14+14=76 generalized symmetries in parastrophic slices.

Eduard I. Vatutin, About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7.
Eduard I. Vatutin, Proving lists (4, 5, 6, 7), Jul 28 2022.
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358394, A358891.

a(n) <= 6*A000041(n)^3. - Eduard I. Vatutin, Dec 29 2022

A358891 Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n in parastrophic slices.

Original entry on oeis.org

6, 0, 0, 76, 44, 0, 145

Offset: 1

Eduard I. Vatutin, Dec 05 2022

An orthogonal diagonal Latin square is a square that has at least one orthogonal diagonal mate.
A diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv are some permutations that describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). A diagonal Latin square A has a generalized symmetry in parastrophic slices if for all cells A[x][y] = v and A[a'][b'] = c' the relation is satisfied: (a',b',c') = R(Px[x],Py[y],Pv[v]), where R is one of 6 possible parastrophic transformations:
   1. (x,y,v) -> (a,b,c) (trivial).
   2. (x,v,y) -> (a,b,c).
   3. (y,x,v) -> (a,b,c) (transpose).
   4. (y,v,x) -> (a,b,c).
   5. (v,x,y) -> (a,b,c).
   6. (v,y,x) -> (a,b,c).
A set of squares with selected parastrophic transformation R forms one of 6 parastrophic slices. Diagonal Latin squares with a generalized symmetry are a special case of generalized symmetries in parastrophic slice # 1. Diagonal Latin squares with generalized symmetries in parastrophic slices are rare; usually they have a large number of transversals, orthogonal mates, etc.
a(8) >= 3874, a(9) >= 8907, a(10) >= 3592.
a(n) <= 6*A000041(n)^3. - Eduard I. Vatutin, Jan 01 2023
For all orders in which orthogonal diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) in first parastrophic slice and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 24 2023, updated Mar 25 2023
The set of generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so A358394(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so a(n) <= A358515(n). - Eduard I. Vatutin, Jan 28 2023

			For order n=4 there are 5 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1},
  2 - {1,1,2},
  3 - {1,3},
  4 - {2,2},
  5 - {4}.
Diagonal Latin squares of order n=4 have a(4)=76 different types of generalized symmetries in parastrophic slices.
Slice 1 (10 generalized symmetries), R=(x,y,v):
  1. A=0123321010322301 (string representation of the square), Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 1-(1,1,1).
  2. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 1-(1,4,4).
  ...
  10. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 1-(5,5,2).
Slice 2 (10 generalized symmetries), R=(x,v,y):
  11. A=0123321010322301, Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3], L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 2-(1,1,1).
  12. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 2-(1,4,4).
  ...
  20. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 2-(5,5,2).
Slice 3 (14 generalized symmetries).
Slice 4 (14 generalized symmetries).
Slice 5 (14 generalized symmetries).
Slice 6 (14 generalized symmetries).
Total 10+10+14+14+14+14=76 generalized symmetries in parastrophic slices.

Eduard I. Vatutin, About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7.
Eduard I. Vatutin, Proving lists (4, 5, 7, 8, 9, 10), Jul 31 2022
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358515, A358394.

cp's OEIS Frontend

A287649 Number of horizontally symmetric diagonal Latin squares of order 2n with the first row in ascending order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A292516 Number of horizontally symmetric diagonal Latin squares of order 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A292517 Number of doubly symmetric diagonal Latin squares of order 4n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A296060 Number of one-plane symmetric diagonal Latin squares of order 2n with first row 0,1,...,2n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A357473 Number of types of generalized symmetries in diagonal Latin squares of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358394 Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358515 Number of types of generalized symmetries in diagonal Latin squares of order n in parastrophic slices.

Views

Author

Keywords

Comments

Examples

Links