A (k1, k2)-design is a type of combinatorial design that is used in the field of design of experiments. It is a mathematical model that specifies the arrangement of experimental units in an experiment.

In a (k1, k2)-design, the experimental units are divided into k1 groups, called blocks, and each block consists of k2 units. The blocks are arranged in such a way that every pair of units from different blocks is included in the same number of blocks. This ensures that the effects of different blocks on the experimental units are evenly distributed, making it easier to determine the effect of each experimental factor on the outcome of the experiment.

One example of a (k1, k2)-design is the Latin square design, which is often used in agricultural experiments. In this design, the experimental units are arranged in a k1 x k2 grid, with each row and each column representing a different block. This ensures that every pair of units from different blocks is included in exactly one block, making it easy to determine the effects of different factors on the outcome of the experiment.

Another example of a (k1, k2)-design is the Balanced Incomplete Block design (BIB), which is commonly used in psychological experiments. In this design, the experimental units are divided into k1 blocks, and each block consists of k2 units. However, unlike the Latin square design, the units within each block are not arranged in a specific pattern. This allows for greater flexibility in the design of the experiment, but it also makes it more difficult to determine the effects of different factors on the outcome of the experiment.

Overall, (k1, k2)-designs are useful for arranging experimental units in a way that allows for easy determination of the effects of different factors on the outcome of the experiment. They are widely used in a variety of fields, including agriculture, psychology, and other areas of science and engineering.