cp's OEIS Frontend

42*a(n) is a maximum number of intercalates in a Latin square of order 2^n-1 (see A092237). - Eduard I. Vatutin, Apr 24 2025

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

Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020

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

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

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

a(16) <= 2, a(17) = 0, a(18) <= 9, a(19) = 0, a(20) <= 1, a(21) <= 11, a(22) <= 9, a(23) = 0, a(24) <= 16, a(25) = 0, a(26) <= 29. - Eduard I. Vatutin, added Sep 10 2023, updated Mar 01 2025

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

0 <= A307163(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020

a(10) >= 109, a(11) >= 172, a(12) >= 324, a(13) >= 180, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 547, a(19) >= 457, a(20) >= 1100, a(21) >= 785, a(22) >= 887, a(23) >= 899, a(24) >= 1680, a(25) >= 1700, a(26) >= 1299, a(27) >= 1372, a(28) >= 2892. - Eduard I. Vatutin, May 31 2021, updated Mar 02 2025

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 05 2025

Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - Omar E. Pol, Nov 13 2008

a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - Álvar Ibeas, Nov 29 2015

a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Ryser conjectured that a(n) >= 1 for all n. For even orders the number is 0, since the group table for Z_2n has no transversals.

a(5)<=814, a(6)<=43093, a(7)<=215721. - Eduard I. Vatutin, added Apr 09 2024, updated Jan 13 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.

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

Table showing minimums and maximums:

order | 4 5 6 7 8 9 10

min number of intercalates | 12 0 - 0 16 0 5

max number of intercalates | 12 0 - 18 112 72 53 (this sequence)

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.

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 (see A383368 and A092237) for some orders N (at least N in {2, 4, 6, 10, 18}).

Pine Latin squares do not exist for odd orders because they must be horizontally symmetric.

Hypothesis: number of transversals in pine Latin squares of all orders N=4k+2 is zero (verified for orders N<=18).

a(9)>=64, a(10)>=103, a(11)>=145, a(12)>=259, a(13)>=200, a(14)>=362, a(15)>=536, a(16)>=794, a(17)>=705, a(18)>=655, a(19)>=469, a(20)>=1362, a(21)>=985, a(22)>=1435, a(23)>=967, a(24)>=1754, a(25)>=1679, a(26)>=2040, a(28)>=2803. - Eduard I. Vatutin, added Aug 13 2024, updated Jun 25 2025

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).

Brown's diagonal Latin squares are special case of plain symmetry diagonal Latin squares that do not exist for odd orders.

a(6)>=252, a(7)>=385, a(8)>=960, a(9)>=329, a(10)>=356, a(11)>=497, a(12)>=1008, a(13)>=497, a(14)>=524.