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%I #29 Mar 25 2023 09:47:40
%S 6,0,0,76,44,0,145
%N Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n in parastrophic slices.
%C An orthogonal diagonal Latin square is a square that has at least one orthogonal diagonal mate.
%C 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:
%C 1. (x,y,v) -> (a,b,c) (trivial).
%C 2. (x,v,y) -> (a,b,c).
%C 3. (y,x,v) -> (a,b,c) (transpose).
%C 4. (y,v,x) -> (a,b,c).
%C 5. (v,x,y) -> (a,b,c).
%C 6. (v,y,x) -> (a,b,c).
%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.
%C a(8) >= 3874, a(9) >= 8907, a(10) >= 3592.
%C a(n) <= 6*A000041(n)^3. - Eduard I. Vatutin, Jan 01 2023
%C 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
%C 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
%C Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so a(n) <= A358515(n). - _Eduard I. Vatutin_, Jan 28 2023
%H 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>.
%H Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/odls_gen_symms/n4_odls_symmetries_list.html">4</a>, <a href="http://evatutin.narod.ru/odls_gen_symms/n5_оdls_symmetries_list.html">5</a>, <a href="http://evatutin.narod.ru/odls_gen_symms/n7_оdls_symmetries_list.html">7</a>, <a href="http://evatutin.narod.ru/odls_gen_symms/n8_odls_symmetries_list.html">8</a>, <a href="http://evatutin.narod.ru/odls_gen_symms/n9_odls_symmetries_list.html">9</a>, <a href="http://evatutin.narod.ru/odls_gen_symms/n10_odls_symmetries_list.html">10</a>), Jul 31 2022
%H <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e For order n=4 there are 5 different multisets L(P) with codes listed below in format "code - multiset":
%e 1 - {1,1,1,1},
%e 2 - {1,1,2},
%e 3 - {1,3},
%e 4 - {2,2},
%e 5 - {4}.
%e Diagonal Latin squares of order n=4 have a(4)=76 different types of generalized symmetries in parastrophic slices.
%e Slice 1 (10 generalized symmetries), R=(x,y,v):
%e 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).
%e 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).
%e ...
%e 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).
%e Slice 2 (10 generalized symmetries), R=(x,v,y):
%e 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).
%e 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).
%e ...
%e 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).
%e Slice 3 (14 generalized symmetries).
%e Slice 4 (14 generalized symmetries).
%e Slice 5 (14 generalized symmetries).
%e Slice 6 (14 generalized symmetries).
%e Total 10+10+14+14+14+14=76 generalized symmetries in parastrophic slices.
%Y Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358515, A358394.
%K nonn,more,hard
%O 1,1
%A _Eduard I. Vatutin_, Dec 05 2022