A293777 -id:A293777 - cp's OEIS frontend

A287650 Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.

2, 12288, 81217160478720, 6101215007109090122576072540160

Offset: 1

Eduard I. Vatutin, May 29 2017

A doubly symmetric square has symmetries in both the 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) <= A293777(4n). - Eduard I. Vatutin, May 26 2021
a(n)/(A001147(n)*2^(n*(4*n-3))) is the number of 2n X 2n grids with two instances of each of 1..n on the main diagonal and in each row and column with the first row in nondescreasing order. - Andrew Howroyd, May 30 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
Reflection of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reflection of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric.
From _Andrew Howroyd_, May 30 2021: (Start)
a(2) = 4*3*1024 = 12288. The 4 base quarter square arrangements are:
  1 1 2 2  1 1 2 2  1 1 2 2  1 1 2 2
  1 2 1 2  1 2 2 1  2 2 1 1  2 2 1 1
  2 1 2 1  2 2 1 1  1 1 2 2  2 2 1 1
  2 2 1 1  2 1 1 2  2 2 1 1  1 1 2 2
(End)

A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, value a(4) is wrong (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian)
E. I. Vatutin, S. E. Kochemazov, 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, 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).
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. A001147, A003191, A287649, A292517, A293777, A340550.

a(n) = A292517(n) / (4n)!.

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

A340546 Number of main classes of diagonal Latin squares of order 2n that contain a one-plane symmetric square.

Original entry on oeis.org

0, 1, 2, 9717

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A one-plane symmetric diagonal Latin square is a vertically or horizontally symmetric diagonal Latin square (see A296060). Such diagonal Latin squares do not exist for odd orders > 1.

			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
Both are one-plane symmetric diagonal Latin squares.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-8 (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. A287649, A287764, A292516, A293777, A293778, A296060, A296061, A340550.

Name clarified by Andrew Howroyd, Oct 22 2023

A293778 Number of centrally symmetric diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 960, 0, 14192640, 5449973760, 118753937326080

Offset: 1

Eduard I. Vatutin, Oct 16 2017

Centrally symmetric diagon 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] (numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist) and 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 A292517(n) <= a(4n). - Eduard I. Vatutin, May 26 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

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (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)
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A292516, A292517, A293777, A340545.

a(n) = A293777(n) * n!.

A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 32, 301, 430090

Offset: 1

Eduard I. Vatutin, Jan 11 2021

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 numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n==2 (mod 4).
Centrally symmetric Latin squares are Latin squares, so a(n) <= A287764(n).
The canonical form (CF) of a square is the lexicographically minimal item within the corresponding main class of diagonal Latin square.
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A340550(n) <= a(n). - Eduard I. Vatutin, May 28 2021

			For n=4 there is a single CF:
  0 1 2 3
  2 3 0 1
  3 2 1 0
  1 0 3 2
so a(4)=1.
For n=5 there are two different CFs:
  0 1 2 3 4   0 1 2 3 4
  2 3 4 0 1   1 3 4 2 0
  4 0 1 2 3   4 2 1 0 3
  1 2 3 4 0   2 0 3 4 1
  3 4 0 1 2   3 4 0 1 2
so a(5)=2.
Example of a centrally symmetric diagonal Latin square of order n=9:
  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

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and 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, and 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)
E. I. Vatutin, About the number of main classes of centrally symmetric diagonal Latin squares of orders 1-9 (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A293777, A293778, A287764, A340550.

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

A287650 Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.

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A340546 Number of main classes of diagonal Latin squares of order 2n that contain a one-plane symmetric square.

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A293778 Number of centrally symmetric diagonal Latin squares of order n.

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A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

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

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

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

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

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