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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 ...
An alternate representation of a Latin square is given by an orthogonal array. For a Latin square of order n this is an n 2 × 3 matrix with columns labeled r, c and s and whose rows correspond to a single position of the Latin square, namely, the row of the position, the column of the position and the symbol in the position. Thus for the order ...
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 ...
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.
Thus, the Cayley table of a group is an example of a latin square. An alternative and more succinct proof follows from the cancellation property . This property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one-to-one map.
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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In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set of symbols (for example, {1,2,...,v}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these ...