In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length ℓ and weight n, such that v = | X | = ( ℓ n ) {\displaystyle v=\left|X\right|={\binom {\ell }{n}}} . Two vectors x, y ∈ X are called ith associates if dist(x, y) = 2i for i = 0, 1, ..., n. The eigenvalues are given by
p i ( k ) = E i ( k ) , {\displaystyle p_{i}\left(k\right)=E_{i}\left(k\right),}
q k ( i ) = μ k v i E i ( k ) , {\displaystyle q_{k}\left(i\right)={\frac {\mu _{k}}{v_{i}}}E_{i}\left(k\right),}
where
μ i = ℓ − 2 i + 1 ℓ − i + 1 ( ℓ i ) , {\displaystyle \mu _{i}={\frac {\ell -2i+1}{\ell -i+1}}{\binom {\ell }{i}},}
and Ek(x) is an Eberlein polynomial defined by
E k ( x ) = ∑ j = 0 k ( − 1 ) j ( x j ) ( n − x k − j ) ( ℓ − n − x k − j ) , k = 0 , … , n . {\displaystyle E_{k}\left(x\right)=\sum _{j=0}^{k}(-1)^{j}{\binom {x}{j}}{\binom {n-x}{k-j}}{\binom {\ell -n-x}{k-j}},\qquad k=0,\ldots ,n.}
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