Fisher's inequality

In combinatorial mathematics, Fisher's inequality, named after Ronald Fisher, is a necessary condition for the existence of a balanced incomplete block design satisfying certain prescribed conditions.

Fisher, a population geneticist and statistician, 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:

* "v" be the number of varieties of plants;
* "b" be the number of blocks.

It was required that:

* "k" different varieties are in each block, "k" < "v"; no variety occurs twice in any one block;
* any two varieties occur together in exactly &lambda; blocks;

Fisher's inequality states simply that

:$v\; ge\; b.,$

References

* R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", "Annals of Eugenics", volume 10, 1940, pages 52&ndash;75.
* R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", "Annals of Mathematical Statistics", 1949, pages 619&ndash;620.