A394485 Number of horizontally symmetric reduced Latin squares of order 2n.

1, 2, 80, 428032

Offset

1,2

Comments

The sum of symmetric elements a[i, j] and a[i, n+1-j] in a horizontally symmetric reduced Latin square of order n is constant and equal to n+1 for all pairs of elements (with rows and columns numbered from 1 to n and element values from 1 to n), based on the definition given in A287649 by Eduard I. Vatutin. Horizontal symmetry is of interest in the Latin stable marriage problem because the even-sized maximal examples given in A351413 have this property, as do the lower bounds preference profiles specified in A005154 and A065982.

Links

Table of n, a(n) for n=1..4.

A. T. Benjamin, C. Converse, and H. A. Krieger, Note. How do I marry thee? Let me count the ways, Discrete Appl. Math. 59 (1995) 285-292 (Section 3 defines vertical reflection relation).

E. G. Thurber, Concerning the maximum number of stable matchings in the stable marriage problem, Discrete Math., 248 (2002), 195-219 (Section 10).

Example

For order 4, the a(2)=2 solutions are:
  1234...1234
  2143...2413
  3412...3142
  4321...4321
For order 6, the example shown in A351413 is:
  123456
  214365
  365214
  456123
  541632
  632541

Prog

(MiniZinc)
int: n = 8;  %reduced Latin squares of even-size order n with horizonal symmetry
% 1, 2, 80, 428032
set of int: Elements = 1..n;
array[Elements, Elements] of var Elements: L;  % the Latin square
include "alldifferent.mzn";
constraint forall(i in Elements) (  % Latin square constraint
  alldifferent([L[i, j] | j in Elements]) /\
  alldifferent([L[j, i] | j in Elements]));
constraint forall(i in Elements) (  % reduced form
  L[1, i] = i /\
  L[i, 1] = i);
constraint forall(r in Elements, i in 1..n div 2) ( % horizontal symmetry
  L[r, i] + L[r, n + 1 - i] = n + 1);
solve satisfy;

Crossrefs

Cf. A394486, A287649, A287650, A351413, A005154, A065982, A336412, A000315.

Sequence in context: A291331 A369468 A293290 * A056972 A051391 A041799

Adjacent sequences: A394482 A394483 A394484 * A394486 A394487 A394488

Keyword

nonn,more

Author

Dan Eilers, Mar 22 2026

Status

approved