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A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. [4] For example, the Latin square above is not reduced because its first column is A, C, B rather than A, B, C. Any Latin square can be reduced by permuting (that is, reordering) the rows and columns ...
A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells ...
Latin squares and finite quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic.The listing below will consider the examples of some very small orders, which is the side length of the square, or the number of elements in the equivalent quasigroup.
An OA(n 2, 3, n, 2) is equivalent to a Latin square of order n. For k ≤ n+1, an OA(n 2, k, n, 2) is equivalent to a set of k − 2 mutually orthogonal Latin squares of order n. Such index one, strength 2 orthogonal arrays are also known as Hyper-Graeco-Latin square designs in the statistical literature.
A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned hyperplane containing it. [ 1 ] When sampling a function of N {\displaystyle N} variables, the range of each variable is divided into M {\displaystyle M} equally probable intervals.
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When k = 1, that is, there is only one row, since the Latin rectangles are normalized there is no choice for what this row can be. The table also shows that L(n − 1, n) = L(n, n), which follows since if only one row is missing, the missing entry in each column can be determined from the Latin square property and the rectangle can be uniquely extended to a Latin square.
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