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; studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.
Let:
To be a balanced incomplete block design it is required that:
Fisher's inequality states simply that

Proof

Let the incidence matrix be a matrix defined so that is 1 if element is in block and 0 otherwise. Then is a matrix such that and for. Since,, so ; on the other hand,, so.

Generalization

Fisher's inequality is valid for more general classes of designs. A pairwise balanced design is a set together with a family of non-empty subsets of such that every pair of distinct elements of is contained in exactly subsets. The set is allowed to be one of the subsets, and if all the subsets are copies of, the PBD is called "trivial". The size of is and the number of subsets in the family is.
Theorem: For any non-trivial PBD,.
This result also generalizes the Erdős–De Bruijn theorem:
For a PBD with having no blocks of size 1 or size,, with equality if and only if the PBD is a projective plane or a near-pencil.
In another direction, Ray-Chaudhuri and Wilson proved in 1975 that in a design, the number of blocks is at least.