A354068 - OEIS

%I #16 Mar 20 2023 21:06:19

%S 1,0,0,4,5,0,8,8,14

%N Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

%C An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate.

%C a(10) <= 60, a(11) <= 279, a(12) <= 588, a(13) <= 9610.

%C Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= a(n) <= A360220(n) <= A287648(n). - _Eduard I. Vatutin_, Mar 03 2023

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1709">About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11</a> (in Russian).

%H E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, <a href="http://evatutin.narod.ru/evatutin_spectra_t_dt_i_o_small_orders_thesis.pdf">On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order</a>, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)

%H Eduard I. Vatutin, <a href="/A354068/a354068.txt">Proving list (best known examples)</a>.

%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.

%e One of the best orthogonal diagonal Latin squares of order n=9

%e   0 1 2 3 4 5 6 7 8

%e   1 2 3 8 6 4 7 0 5

%e   5 4 6 0 7 8 3 1 2

%e   7 3 1 5 2 6 0 8 4

%e   8 7 4 6 1 2 5 3 0

%e   3 0 5 4 8 7 1 2 6

%e   4 6 7 2 3 0 8 5 1

%e   6 5 8 1 0 3 2 4 7

%e   2 8 0 7 5 1 4 6 3

%e has orthogonal diagonal mate

%e   0 1 2 3 4 5 6 7 8

%e   2 3 8 7 5 6 4 1 0

%e   1 5 4 8 6 0 2 3 7

%e   8 7 0 6 1 3 5 4 2

%e   5 0 1 2 7 8 3 6 4

%e   4 6 7 0 3 2 8 5 1

%e   3 8 5 4 0 7 1 2 6

%e   7 4 6 5 2 1 0 8 3

%e   6 2 3 1 8 4 7 0 5

%e and 14 diagonal transversals, which is the minimal number, so a(9)=14.

%Y Cf. A287647, A287648, A345370, A349199, A360220.

%K nonn,more,hard

%O 1,4

%A _Eduard I. Vatutin_, May 16 2022