In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular

association scheme The theory of association schemes arose in statistics, in the theory of design of experiments, experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatori ...

. It consists of the set of all binary vectors ''X'' of length ''ℓ'' and weight ''n'', such that

v=\left, X\=\binom

.F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978. Two vectors ''x'', ''y'' ∈ ''X'' are called ''i''th associates if dist(''x'', ''y'') = 2''i'' for ''i'' = 0, 1, ..., ''n''. The

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

are given by :

p_\left(k\right)=E_\left(k\right),

q_\left(i\right)=\fracE_\left(k\right),

where :

\mu_=\frac\binom,

and ''E''_''k''(''x'') is an Eberlein polynomial defined by :

E_\left(x\right)=\sum_^(-1)^\binom \binom\binom,\qquad k=0,\ldots,n.

References

Combinatorics {{combin-stub