A383570 -id:A383570 - cp's OEIS frontend

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

1, 12, 27, 80, 125, 252, 343, 576, 729, 1100, 1331, 1872, 2197, 2940, 3375, 4352, 4913, 6156, 6859, 8400, 9261, 11132, 12167, 14400, 15625

Offset: 1

Eduard I. Vatutin, Apr 24 2025

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.
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.
All pine Latin squares are horizontally symmetric column-inverse Latin squares.
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.).
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}).
Pine Latin squares do not exist for odd orders due to they are horizontally symmetric.
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.

			For order N=8 pine Latin square
  0 1 2 3 4 5 6 7
  1 2 3 0 7 4 5 6
  2 3 0 1 6 7 4 5
  3 0 1 2 5 6 7 4
  4 5 6 7 0 1 2 3
  5 6 7 4 3 0 1 2
  6 7 4 5 2 3 0 1
  7 4 5 6 1 2 3 0
have 80 intercalates.
.
For order N=10 pine Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 3 4 0 9 5 6 7 8
  2 3 4 0 1 8 9 5 6 7
  3 4 0 1 2 7 8 9 5 6
  4 0 1 2 3 6 7 8 9 5
  5 6 7 8 9 0 1 2 3 4
  6 7 8 9 5 4 0 1 2 3
  7 8 9 5 6 3 4 0 1 2
  8 9 5 6 7 2 3 4 0 1
  9 5 6 7 8 1 2 3 4 0
have 125 intercalates.
.
For order N=12 pine Latin square
  0 1 2 3 4 5 6 7 8 9 10 11
  1 2 3 4 5 0 11 6 7 8 9 10
  2 3 4 5 0 1 10 11 6 7 8 9
  3 4 5 0 1 2 9 10 11 6 7 8
  4 5 0 1 2 3 8 9 10 11 6 7
  5 0 1 2 3 4 7 8 9 10 11 6
  6 7 8 9 10 11 0 1 2 3 4 5
  7 8 9 10 11 6 5 0 1 2 3 4
  8 9 10 11 6 7 4 5 0 1 2 3
  9 10 11 6 7 8 3 4 5 0 1 2
  10 11 6 7 8 9 2 3 4 5 0 1
  11 6 7 8 9 10 1 2 3 4 5 0
have 252 intercalates.

R. Bean, Critical sets in Latin squares and associated structures, Ph.D. Thesis, The University of Queensland, 2001.
Eduard I. Vatutin, About the properties of pine Latin squares (in Russian).
Eduard I. Vatutin, Proving list (examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A002860, A016755, A089207, A092237, A099721, A338522, A383570.

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).

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).

cp's OEIS Frontend

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

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