cp's OEIS Frontend

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

%I A357473 #41 May 03 2023 23:29:06
%S A357473 1,0,0,10,8,12,12
%N A357473 Number of types of generalized symmetries in diagonal Latin squares of order n.
%C A357473 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.
%C A357473 a(n) <= A000041(n)^3. - _Eduard I. Vatutin_, Dec 29 2022
%C A357473 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
%C A357473 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
%C A357473 Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so A358394(n) <= a(n). - _Eduard I. Vatutin_, Jan 25 2023
%H A357473 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2058">About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7</a>, 28 Jul 2022.
%H A357473 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/dls_gen_symms/n4_dls_symmetries_list.html">4</a>, <a href="http://evatutin.narod.ru/dls_gen_symms/n5_dls_symmetries_list.html">5</a>, <a href="http://evatutin.narod.ru/dls_gen_symms/n6_dls_symmetries_list.html">6</a>, <a href="http://evatutin.narod.ru/dls_gen_symms/n7_dls_symmetries_list.html">7</a>), Jul 28 2022.
%H A357473 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A357473 For order n=5 there are 7 different multisets L(P) with codes listed below in format "code - multiset":
%e A357473   1 - {1,1,1,1,1},
%e A357473   2 - {1,1,1,2},
%e A357473   3 - {1,1,3},
%e A357473   4 - {1,2,2},
%e A357473   5 - {1,4},
%e A357473   6 - {2,3},
%e A357473   7 - {5}.
%e A357473 Diagonal Latin squares of order n=5 has a(5)=8 different types of generalized symmetries:
%e A357473 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).
%e A357473 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).
%e A357473 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).
%e A357473 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).
%e A357473 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).
%e A357473 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).
%e A357473 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).
%e A357473 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).
%Y A357473 Cf. A000041, A274171, A287649, A287650, A293777, A358515, A358394, A358891.
%K A357473 nonn,more,hard
%O A357473 1,4
%A A357473 _Eduard I. Vatutin_, Sep 29 2022