A307163 - OEIS

%I #65 Oct 20 2024 13:57:15

%S 0,0,0,12,0,9,0,0,0,0,0,0,0,0,0

%N Minimum number of intercalates in a diagonal Latin square of order n.

%C An intercalate is a 2 X 2 subsquare of a Latin square.

%C Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A307164(n) <= A092237(n). - _Eduard I. Vatutin_, Sep 21 2020

%C Every intercalate is a partial loop and every partial loop is a loop, so 0 <= a(n) <= A307170(n) <= A307166(n). - _Eduard I. Vatutin_, Oct 19 2020

%C a(n)=0 for all orders n for which cyclic diagonal Latin squares exist (see A007310) due to all cyclic diagonal Latin squares don't have intercalates. - _Eduard I. Vatutin_, Aug 07 2023

%C a(n)=0 for all orders n for which diagonalized cyclic diagonal Latin squares exist (see A372922) due to all diagonalized cyclic diagonal Latin squares don't have intercalates. - _Eduard I. Vatutin_, Sep 24 2024

%C a(16) <= 5, a(17) = 0, a(18) <= 10, a(19) = 0, a(20) <= 3, a(21) <= 11, a(22) <= 9, a(23) = 0, a(24) <= 16, a(25) = 0. - _Eduard I. Vatutin_, added Sep 10 2023, updated Sep 24 2024

%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&amp;m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian)

%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1333">About the minimum number of intercalates in a diagonal Latin squares of order 9</a> (in Russian)

%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1403">On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops</a> (in Russian)

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2475">About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14</a> (in Russian)

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2476">About the minimum number of intercalates in diagonal Latin squares of order 15</a> (in Russian)

%H E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, <a href="https://doi.org/10.1007/978-3-030-66895-2_9">Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10</a>, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.

%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="/A307163/a307163_2.txt">Proving list (best known examples)</a>.

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

%Y Cf. A007310, A307164, A092237, A307170, A307166, A345760, A372922.

%K nonn,more,hard

%O 1,4

%A _Eduard I. Vatutin_, Mar 27 2019

%E a(9) added by _Eduard I. Vatutin_, Sep 21 2020

%E a(10)-a(13) added by _Eduard I. Vatutin_, Apr 01 2021

%E a(14)-a(15) added by _Eduard I. Vatutin_, Sep 24 2024