How did Ronald Fisher and Frank Yates connect experimental design to combinatorics?

Sir Ronald Fisher and Frank Yates were instrumental in bridging the gap between combinatorics and the statistical design of experiments. While Fisher pioneered the principles of randomization and analysis of variance to eliminate bias and evaluate multiple factors simultaneously, he and Yates utilized specific combinatorial structures to organize these experiments efficiently.1

A primary example of this connection is the Latin square, a square array of $n$ elements where each element appears exactly once in every row and column. Fisher and Yates employed these arrangements in agricultural and plant-breeding research to control for environmental variability across a field. By assigning different treatments to plots according to the constraints of a Latin square, they ensured that each treatment was balanced across rows and columns, allowing researchers to isolate the effects of specific variables while minimizing experimental error. Their work demonstrated that complex combinatorial designs are essential for extracting reliable scientific inferences from limited experimental resources.21