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

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

Offset: 1

Author

Eduard I. Vatutin, Sep 29 2022

Keywords

• nonn
• more
• hard

Comments

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

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}.
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).
		

Links

• 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.

Crossrefs

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