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

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

Offset: 1

Author

Eduard I. Vatutin, Nov 20 2022

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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).
		

Links

• 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.

Crossrefs

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