A **Latin square** is an *n* × *n* grid with a number from 1 to *n* in each cell, such that no number appears twice in a row or twice in a column.

It’s not obvious that Latin squares exist for all *n*, but they do, and in fact there are a lot of them. The exact number is known only for *n* ≤ 11. See [1] for the known values. There are upper and lower bounds for all *n*, and this post will look at how good the bounds are.

A particularly simple result is that number of Latin squares of size *n* is bounded below by the superfactorial of *n* [2]. That is, if *L*_{n} is the number of Latin squares of size *n* and the superfactorial of *n* is defined by

*S*(*n*) = 1! × 2! × 3! × … × *n*!

then

*L*_{n} ≥ *S*(*n*).

A **reduced Latin square** is a Latin square in which the elements of the first row and first column are in numerical order. The image at the top of the post is a reduced Latin square. If *R*_{n} is the number of reduced Latin squares of size *n* then

*L*_{n} = *n*! (*n*-1)! *R*_{n}

and so we can divide the lower bound on *S*_{n} by *n*! (*n*-1)! to get a lower bound on *R*_{n}:

*R*_{n} ≥ *S*(*n*-2).

Ronald Alter [2] gives the following upper bound on *R*_{n}:

Here’s now the lower bound, exact value, and upper bound compare for *n* up to 6.

|---+-------+-------+---------| | n | lower | exact | upper | |---+-------+-------+---------| | 1 | 1 | 1 | 1 | | 2 | 1 | 1 | 1 | | 3 | 1 | 1 | 2 | | 4 | 2 | 4 | 24 | | 5 | 12 | 56 | 3456 | | 6 | 288 | 9408 | 9553280 | |---+-------+-------+---------|

The numbers get very big for larger *n*. so let’s switch over to a log scale.

Here’s a plot corresponding to the logarithms of the values above, including all known values of *R*_{n}.

The lower and upper bounds are remarkably symmetric about the exact value, which suggests that their average would make a good estimate. Let’s look at a plot.

Indeed, the average of the logs of the bounds is very close to the log of the actual value. This says the number of reduced Latin squares of size *n* is approximately the geometric mean of its upper and lower bounds, at least for *n* up to 11.

We can factor *S*(*n*-2) out of the upper bound on *R*_{n } when computing the geometric mean, and this gives us the approximation

## References

[1] OEIS A000315

[2] Ronald Alter. The American Mathematical Monthly. Vol. 82, No. 6 (Jun. – Jul., 1975), pp. 632-634