An Latin square is a Latin
rectangle with .
Specifically, a Latin square consists of sets of the numbers 1 to arranged in such a way that no orthogonal (row or column)
contains the same number twice. For example, the two Latin squares of order two are
given by
(1)
the 12 Latin squares of order three are given by
(2)
and two of the 576 Latin squares of order 4 are given by
The numbers
of Latin squares of order ,
2, ... are 1, 2, 12, 576, 161280, ... (OEIS A002860).
The number
of isotopically distinct Latin squares of order , 2, ... are 1, 1, 1, 2, 2, 22, 564, 1676267, ... (OEIS A040082).
A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct.
For example, the two Latin squares
(4)
are orthogonal. The number of pairs of orthogonal Latin squares of order , 2, ... are 0, 0, 36, 3456, ... (OEIS A072377).
The number of Latin squares of order with first row given by is the same as the number of fixed diagonal Latin
squares of order
(i.e., the number of Latin squares of order having all 1s along their main diagonals). For , 2, ..., the numbers of such matrices are 1, 1, 2, 24, 1344,
1128960, ... (OEIS A000479) and the total number
of Latin squares of order
is equal to this number times .
A normalized, or reduced, Latin square is a Latin square with the first row and column given by .
General formulas for the number of normalized Latin squares are given by Nechvatal (1981), Gessel (1987), and Shao
and Wei (1992), but the asymptotic value of is not known (MacKay and Wanless 2005). The total
number of Latin squares
of order
can then be computed from
(5)
The numbers of normalized Latin squares of order , 2, ..., are summarized in the following table (OEIS A000315).
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