Fisher's inequality

Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.

Let:

• v be the number of varieties of plants;
• b be the number of blocks.

To be a balanced incomplete block design it is required that:

• k different varieties are in each block, 1 ≤ k < v; no variety occurs twice in any one block;
• any two varieties occur together in exactly λ blocks;
• each variety occurs in exactly r blocks.

Fisher's inequality states simply that

b ≥ v.