Fisher's inequality

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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 λ blocks;
each variety occurs in exactly r blocks.

Fisher's inequality states simply that

$b\geq v.\,$

Proof

Let the incidence matrix M be a v×b matrix defined so that M_i,j is 1 if element i is in block j and 0 otherwise. Then B=MM^T is a v×v matrix such that B_i,i = r and B_i,j = λ for i ≠ j. Since r ≠ λ, det(B) ≠ 0, so rank(B) = v; on the other hand, rank(B) ≤ rank(M) ≤ b, so v ≤ b.

Generalization

Fisher's inequality remains valid for more general classes of designs.

A pairwise balanced design (or PBD) is a set X together with a family of subsets of X (which need not have the same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly λ (a positive integer) subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X, th PBD is called trivial. The size of X is v and the number of subsets in the family (counted with multiplicity) is b.

Theorem: For any non-trivial PBD, v ≤ b.^[1]

This result also generalizes the famous Erdős-De Bruijn theorem:

For a PBD with λ = 1 having no blocks of size 1 or size v, v ≤ b, with equality if and only if the PBD is a projective plane or a near-pencil.^[2]

Notes

↑ Stinson 2003, pg.193
↑ Stinson 2003, pg.183

References

R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", Annals of Mathematical Statistics, 1949, pages 619–620.

R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, volume 10, 1940, pages 52–75.

Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN 0-387-95487-2

Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

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