cp's OEIS Frontend

A383570 Number of transversals in pine Latin squares of order 4n.

%I A383570 #20 Jul 20 2025 08:53:30
%S A383570 8,384,76032,62881792
%N A383570 Number of transversals in pine Latin squares of order 4n.
%C A383570 A pine Latin square is a not necessarily canonical composite Latin square of order N=2*K formed from specially arranged cyclic Latin squares of order K.
%C A383570 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 A383570 All pine Latin squares are horizontally symmetric column-inverse Latin squares.
%C A383570 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 A383570 Pine Latin squares have interesting properties, for example, maximum known number of intercalates (see A383368 and A092237) for some orders N (at least N in {2, 4, 6, 10, 18}).
%C A383570 Pine Latin squares do not exist for odd orders because they must be horizontally symmetric.
%C A383570 Hypothesis: number of transversals in pine Latin squares of all orders N=4k+2 is zero (verified for orders N<=18).
%H A383570 Richard 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 A383570 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2995">About the properties of pine Latin squares</a> (in Russian).
%H A383570 Eduard I. Vatutin, <a href="/A383570/a383570.txt">Proving list (examples)</a>.
%H A383570 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A383570 For order N=8 pine Latin square
%e A383570   0 1 2 3 4 5 6 7
%e A383570   1 2 3 0 7 4 5 6
%e A383570   2 3 0 1 6 7 4 5
%e A383570   3 0 1 2 5 6 7 4
%e A383570   4 5 6 7 0 1 2 3
%e A383570   5 6 7 4 3 0 1 2
%e A383570   6 7 4 5 2 3 0 1
%e A383570   7 4 5 6 1 2 3 0
%e A383570 has 384 transversals.
%e A383570 .
%e A383570 For order N=10 pine Latin square
%e A383570   0 1 2 3 4 5 6 7 8 9
%e A383570   1 2 3 4 0 9 5 6 7 8
%e A383570   2 3 4 0 1 8 9 5 6 7
%e A383570   3 4 0 1 2 7 8 9 5 6
%e A383570   4 0 1 2 3 6 7 8 9 5
%e A383570   5 6 7 8 9 0 1 2 3 4
%e A383570   6 7 8 9 5 4 0 1 2 3
%e A383570   7 8 9 5 6 3 4 0 1 2
%e A383570   8 9 5 6 7 2 3 4 0 1
%e A383570   9 5 6 7 8 1 2 3 4 0
%e A383570 has no transversals.
%e A383570 .
%e A383570 For order N=12 pine Latin square
%e A383570   0 1 2 3 4 5 6 7 8 9 10 11
%e A383570   1 2 3 4 5 0 11 6 7 8 9 10
%e A383570   2 3 4 5 0 1 10 11 6 7 8 9
%e A383570   3 4 5 0 1 2 9 10 11 6 7 8
%e A383570   4 5 0 1 2 3 8 9 10 11 6 7
%e A383570   5 0 1 2 3 4 7 8 9 10 11 6
%e A383570   6 7 8 9 10 11 0 1 2 3 4 5
%e A383570   7 8 9 10 11 6 5 0 1 2 3 4
%e A383570   8 9 10 11 6 7 4 5 0 1 2 3
%e A383570   9 10 11 6 7 8 3 4 5 0 1 2
%e A383570   10 11 6 7 8 9 2 3 4 5 0 1
%e A383570   11 6 7 8 9 10 1 2 3 4 5 0
%e A383570 has 76032 transversals.
%Y A383570 Cf. A002860, A091323, A090741, A338522, A383368.
%K A383570 nonn,more,hard
%O A383570 1,1
%A A383570 _Eduard I. Vatutin_, Apr 30 2025