A383673 a(n) is the number of n X n Latin squares obeying a certain self-referential property defined in the comments.

1, 2, 0, 1, 2, 0, 0, 2, 2, 0, 4, 0, 4, 0, 0, 2

Offset

1,2

Comments

a(n) = |B_n| where B_n is the set defined as follows:

Let G be a grid of n X n cells. Then, B_n is the set of all G which satisfy these properties:

0. Each cell g(r, c) of G contains exactly one element from Z_n.

1. G is a Latin square.

Two cells, A = g(a_r, a_c) and B = g(b_r, b_c), form a referential domino iff a_r = g(A, B).

2. If two horizontally adjacent cells A and B are not in the last column, then (A, B) must be a referential domino. Alternatively stated, for all a_r in Z_n and a_c in Z_{n-2}, given the horizontal domino A = g(a_r, a_c) and B = g(a_r, a_c + 1), then g(A, B) = a_r.

Altering the conditions to index the grid from 1 (instead of 0), and filling with integers 1 to n inclusive does not change the number of grids in B_n, nor does it qualitatively change those grids; they are only transformed by a simple relabeling.

a(1) = 1, and a(2) = 2 may be considered degenerate since they are not restricted by property 2.

It appears that if a(n) is nonzero, then n is a prime power.

It appears that n=3 and n=7 are the only prime powers for which a(n) = 0.

a(n) appears to always be even, except a(1) = a(4) = 1.

Links

Table of n, a(n) for n=1..16.

Jack Cats, Self-Referential Grids, Section 1-3.

Example

The following is an example of G in B_5:
  ..01234
  .......
  0.14023
  1.32410
  2.20341
  3.43102
  4.01234
Properties 0 and 1 are trivially verifiable.
For property 2, the pair A = g(4,1), B = g(4,2):
    is a horizontal domino in row r=4, with neither cell in the last column
    is a referential domino, since g(A, B) = g(1, 2) = 4 = r
The same holds for the rest of the dominoes in G.
G' is a grid obtained by swapping  3 <-> 1 and 2 <-> 0 for every cell in G.
G' is also in B_5.
G and G' are the only 2 members of B_5, thus a(5) = 2.

Crossrefs

Sequence in context: A261249 A058650 A112177 * A115723 A238160 A178524

Adjacent sequences: A383670 A383671 A383672 * A383674 A383675 A383676

Keyword

nonn,more,hard

Author

Luis Novoa, May 04 2025

Status

approved