This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A092237 #43 Jul 06 2025 00:49:30 %S A092237 0,1,0,12,4,27,42,112,72 %N A092237 Maximum number of intercalates in a Latin square of order n. %C A092237 An intercalate is a 2 X 2 subsquare of a Latin square. a(10) >= 125, a(11) >= 172, a(12) >= 324. %C A092237 a(13) >= 208, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 729, a(19) >= 472, a(20) >= 1500, a(21) >= 884, a(22) >= 1497, a(23) >= 983, a(24) >= 1872, a(25) >= 1700, a(26) >= 2197, a(27) >= 648, a(28) >= 2940. - _Eduard I. Vatutin_, Mar 02 2025 %C A092237 If, in theory, all unordered pairs of rows and columns form intercalate in their intersection, total number of intercalates will be (n*(n-1))^2, so a(n) <= (n*(n-1))^2, a(n) is asymptotically less than O(n^4). In practice a(n) << (n*(n-1))^2. - _Eduard I. Vatutin_, Mar 11 2025 %C A092237 a(2^n-1) = 42*A006096(n) for n > 2. - _Eduard I. Vatutin_, Apr 23 2025 %C A092237 Due to existence of the pine Latin squares for even orders N=2n, a(2n) >= A383368(n). Pine Latin squares exist for all even orders, so a(N) >= (2k)^2 * (2k + 1) for N=4k and a(N) >= (2k+1)^3 for N=4k+2. Therefore, asymptotically maximum number of intercalates in Latin squares of even orders N greater or equal than o(k1*N^3), where k1 = 1/8. - _Eduard I. Vatutin_, Apr 30 2025 %C A092237 From _Eduard I. Vatutin_, Jul 01 2025: (Start) %C A092237 Table showing minimums and maximums: %C A092237 order | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 %C A092237 min number of intercalates | 0 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %C A092237 max number of intercalates | 0 1 0 12 4 27 42 112 72 >=125 >=172 >=324 >=208 >=391 >=630 >=960 >=736 >=729 >=472 (this sequence) %C A092237 . %C A092237 order | 20 21 22 23 24 25 26 27 28 29 %C A092237 min number of intercalates | 0 0 0 0 0 0 <=15 0 <=1 0 %C A092237 max number of intercalates | >=1500 >=884 >=1497 >=983 >=1872 >=1700 >=2197 >=648 >=2940 ? (this sequence) %C A092237 (End) %D A092237 I. Wanless, Private communication, 2003. %H A092237 R. Bean, <a href="https://www.researchgate.net/publication/2416446_Critical_Sets_in_Latin_Squares_and_Associated_Structures">Critical sets in Latin squares and associated structures</a>, Ph.D. Thesis, The University of Queensland, 2001. %H A092237 K. Heinrich and W. Wallis, <a href="https://doi.org/10.1007/BFb0091822">The maximum number of intercalates in a Latin square</a>, Combinatorial Math. VIII, Proc. 8th Australian Conf. Combinatorics, 1980, 221-233. %H A092237 Eduard I. Vatutin, <a href="/A092237/a092237_1.txt">Proving list, minimum number of intercalates (best known examples)</a>. %H A092237 Eduard I. Vatutin, <a href="/A092237/a092237.txt">Proving list, maximum number of intercalates (best known examples)</a>. %H A092237 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_3015">About interconnection between maximum number of intercalates in Latin squares of order N=2^n-1 and Gaussian binomial coefficients [n,3] for q=2</a> (in Russian). %H A092237 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %F A092237 If n is a power of 2, a(n) = n^2*(n-1)/4 = A016152(log2(n)); if n is one less than a power of 2, a(n) = n*(n-1)*(n-3)/4 = A006096(log2(n+1))*42. - updated by _Eduard I. Vatutin_, Jun 28 2025 %Y A092237 Cf. A006096, A016152, A091323, A090741, A383368. %K A092237 hard,nonn,more %O A092237 1,4 %A A092237 _Richard Bean_, Feb 17 2004