cp's OEIS Frontend

A383368 Number of intercalates in pine Latin squares of order 2n.

%I A383368 #9 Aug 21 2025 00:14:56
%S A383368 1,12,27,80,125,252,343,576,729,1100,1331,1872,2197,2940,3375,4352,
%T A383368 4913,6156,6859,8400,9261,11132,12167,14400,15625
%N A383368 Number of intercalates in pine Latin squares of order 2n.
%C A383368 Pine Latin square is a none canonical composite Latin square of order N=2*K formed from specially arranged cyclic Latin squares of order K.
%C A383368 By construction, pine Latin square is determined one-to-one by the cyclic square used, so number of pine Latin squares of order N is equal to number of cyclic Latin squares of order N/2.
%C A383368 All pine Latin squares are horizontally symmetric column-inverse Latin squares.
%C A383368 All pine Latin squares for selected order N are isomorphic one to another as Latin squares, so they have same properties (number of transversals, intercalates, etc.).
%C A383368 Pine Latin squares have interesting properties, for example, maximum known number of intercalates for some orders N (at least N in {2, 4, 6, 10, 18}).
%C A383368 Pine Latin squares do not exist for odd orders due to they are horizontally symmetric.
%C A383368 Pine Latin squares of order N=2n exists for all even orders due to existing of corresponding cyclic Latin squares of order n. According to this, maximum number of intercalates in a Latin square A092237(N) >= (2k)^2 * (2k + 1) for N=4k and A092237(N) >= (2k+1)^3 for N=4k+2. Therefore, asimptotically maximum number of intercalates in Latin squares of even orders N greater or equal than o(k1*N^3), where k1 = 1/8.
%H A383368 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 A383368 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2995">About the properties of pine Latin squares</a> (in Russian).
%H A383368 Eduard I. Vatutin, <a href="/A383368/a383368.txt">Proving list (examples)</a>.
%H A383368 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F A383368 Hypothesis: For all known pine Latin squares of orders N=4k+2 number of intercalates a(n) = a(N/2)= a(2k+1) = (N/2)^3 = (2k+1)^3 = A016755((n-1)/2) (verified for N<29).
%F A383368 Hypothesis: For all known pine Latin squares of orders N=4k number of intercalates a(n) = a(N/2) = a(2k) = (N/2)^2 + (N/2)^3 = 4*k^2 + 8*k^3 = (2k)^2 * (2k+1) = 2*A089207(n/2) = 4*A099721(n/2) (verified for N<29).
%e A383368 For order N=8 pine Latin square
%e A383368   0 1 2 3 4 5 6 7
%e A383368   1 2 3 0 7 4 5 6
%e A383368   2 3 0 1 6 7 4 5
%e A383368   3 0 1 2 5 6 7 4
%e A383368   4 5 6 7 0 1 2 3
%e A383368   5 6 7 4 3 0 1 2
%e A383368   6 7 4 5 2 3 0 1
%e A383368   7 4 5 6 1 2 3 0
%e A383368 have 80 intercalates.
%e A383368 .
%e A383368 For order N=10 pine Latin square
%e A383368   0 1 2 3 4 5 6 7 8 9
%e A383368   1 2 3 4 0 9 5 6 7 8
%e A383368   2 3 4 0 1 8 9 5 6 7
%e A383368   3 4 0 1 2 7 8 9 5 6
%e A383368   4 0 1 2 3 6 7 8 9 5
%e A383368   5 6 7 8 9 0 1 2 3 4
%e A383368   6 7 8 9 5 4 0 1 2 3
%e A383368   7 8 9 5 6 3 4 0 1 2
%e A383368   8 9 5 6 7 2 3 4 0 1
%e A383368   9 5 6 7 8 1 2 3 4 0
%e A383368 have 125 intercalates.
%e A383368 .
%e A383368 For order N=12 pine Latin square
%e A383368   0 1 2 3 4 5 6 7 8 9 10 11
%e A383368   1 2 3 4 5 0 11 6 7 8 9 10
%e A383368   2 3 4 5 0 1 10 11 6 7 8 9
%e A383368   3 4 5 0 1 2 9 10 11 6 7 8
%e A383368   4 5 0 1 2 3 8 9 10 11 6 7
%e A383368   5 0 1 2 3 4 7 8 9 10 11 6
%e A383368   6 7 8 9 10 11 0 1 2 3 4 5
%e A383368   7 8 9 10 11 6 5 0 1 2 3 4
%e A383368   8 9 10 11 6 7 4 5 0 1 2 3
%e A383368   9 10 11 6 7 8 3 4 5 0 1 2
%e A383368   10 11 6 7 8 9 2 3 4 5 0 1
%e A383368   11 6 7 8 9 10 1 2 3 4 5 0
%e A383368 have 252 intercalates.
%Y A383368 Cf. A002860, A016755, A089207, A092237, A099721, A338522, A383570.
%K A383368 nonn,easy,changed
%O A383368 1,2
%A A383368 _Eduard I. Vatutin_, Apr 24 2025