Page 53 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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iusz Meszka: Combinatorial Designs 41
A latin square L of side n is commutative (or symmetric) if L(i , j ) = L(j , i ) for all 1 ≤
i , j ≤ n. L is idempotent if L(i , i ) = i for all 1 ≤ i ≤ n. A latin square L of even order
n = 2k is half-idempotent if L (i , i ) = i and L (k + i , k + i ) = i for all 1 ≤ i ≤ k .
The existence of a latin square of order n is equivalent to the existence of a one-
factorization of the complete bipartite graph Kn,n . Moreover, the existence of a com-
mutative idempotent latin square of order n is equivalent to the existence of a one-
factorization of the complete graph Kn .
A latin square is in standard form (or normalized) if both its first column and first row
contain consecutive symbols in an increasing order.
Two latin squares, L and L , of order n are isotopic (or equivalent) if there are three
bijections from the rows, columns and symbols of L to the rows, columns and symbols,
respectively, of L , that map L to L . Latin squares L and L are isomorphic if there exists
a bijection ϕ : S → S such that ϕ(L(i , j )) = L (ϕ(i ), ϕ(j )) for every i , j ∈ S, where S is not
only the set of symbols of each square but also the indexing set for the rows and columns
of each square.
Latin squares are completely enumerated for small orders.
n number of non-isomorphic latin squares number of distinct latin squares
2 11
3 12
4 2 24
5 2 1, 334
6 17 1, 128, 960
7 324 12, 198, 297, 600
8 842, 227 2, 697, 818, 265, 354, 240
9 15, 224, 734, 061, 278, 915, 461, 120
10 2, 750, 892, 211, 809, 148, 994, 633, 229, 926, 400
11 19, 464, 657, 391, 668, 924, 966, 616, 671, 344, 752, 852, 992, 000
Two latin squares, L and L , of order n are orthogonal if the n 2 ordered pairs (L(i , j ),
L (i , j )) are all distinct. A set of latin squares L 1, L 2, . . . , L m is mutually orthogonal (or a
set of MOLS(n )) if for every 1 ≤ i < j ≤ m , L i and L j are orthogonal.
Example 10. A set of three MOLS(4):
1234 1234 1234
4321 3412 2143
2143 4321 3412
3412 2143 4321
In any latin square belonging to some set of MOLS(n), relabeling symbols does not
affect to the orthogonality.
Theorem 7. A pair of orthogonal latin squares of order n exists for all n other than 2 and
6 (for which no such pair exists).
A latin square L of side n is commutative (or symmetric) if L(i , j ) = L(j , i ) for all 1 ≤
i , j ≤ n. L is idempotent if L(i , i ) = i for all 1 ≤ i ≤ n. A latin square L of even order
n = 2k is half-idempotent if L (i , i ) = i and L (k + i , k + i ) = i for all 1 ≤ i ≤ k .
The existence of a latin square of order n is equivalent to the existence of a one-
factorization of the complete bipartite graph Kn,n . Moreover, the existence of a com-
mutative idempotent latin square of order n is equivalent to the existence of a one-
factorization of the complete graph Kn .
A latin square is in standard form (or normalized) if both its first column and first row
contain consecutive symbols in an increasing order.
Two latin squares, L and L , of order n are isotopic (or equivalent) if there are three
bijections from the rows, columns and symbols of L to the rows, columns and symbols,
respectively, of L , that map L to L . Latin squares L and L are isomorphic if there exists
a bijection ϕ : S → S such that ϕ(L(i , j )) = L (ϕ(i ), ϕ(j )) for every i , j ∈ S, where S is not
only the set of symbols of each square but also the indexing set for the rows and columns
of each square.
Latin squares are completely enumerated for small orders.
n number of non-isomorphic latin squares number of distinct latin squares
2 11
3 12
4 2 24
5 2 1, 334
6 17 1, 128, 960
7 324 12, 198, 297, 600
8 842, 227 2, 697, 818, 265, 354, 240
9 15, 224, 734, 061, 278, 915, 461, 120
10 2, 750, 892, 211, 809, 148, 994, 633, 229, 926, 400
11 19, 464, 657, 391, 668, 924, 966, 616, 671, 344, 752, 852, 992, 000
Two latin squares, L and L , of order n are orthogonal if the n 2 ordered pairs (L(i , j ),
L (i , j )) are all distinct. A set of latin squares L 1, L 2, . . . , L m is mutually orthogonal (or a
set of MOLS(n )) if for every 1 ≤ i < j ≤ m , L i and L j are orthogonal.
Example 10. A set of three MOLS(4):
1234 1234 1234
4321 3412 2143
2143 4321 3412
3412 2143 4321
In any latin square belonging to some set of MOLS(n), relabeling symbols does not
affect to the orthogonality.
Theorem 7. A pair of orthogonal latin squares of order n exists for all n other than 2 and
6 (for which no such pair exists).
