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  "body": "\n## Definition\n\nLet $X$ be a nonempty [[finite set|070R982G4B]] and let $d \\ge 1$.\nAn *association scheme* with $d$ classes on $X$ is a [[partition|08QMKTWF9T]] of $X \\times X$ into nonempty relations $R_0, R_1, \\dots, R_d$ such that:\n\n1. $R_0 = \\set{(x, x) : x \\in X}$.\n2. For each $i$, the relation $R_i^T = \\set{(y, x) : (x, y) \\in R_i}$ is equal to $R_{i'}$ for some $i'$.\n3. For all $i, j, k$ there is an integer $p^k_{ij}$ such that for every $(x, y) \\in R_k$ the number of $z \\in X$ with $(x, z) \\in R_i$ and $(z, y) \\in R_j$ equals $p^k_{ij}$.\n\nThe scheme is *symmetric* if $R_i^T = R_i$ for every $i$, and *commutative* if $p^k_{ij} = p^k_{ji}$ for all $i, j, k$.\n\n## Remarks\n\nMany authors, following Bose and Mesner, require the scheme to be symmetric in the definition.\nIn terms of the [[adjacency matrices|08B9CX109G]] $A_i$ of the relations, clause 3 says $A_i A_j = \\sum_k p^k_{ij} A_k$.\nSo the $A_i$ span an algebra of matrices, the Bose–Mesner algebra, which is commutative exactly when the scheme is commutative.\n\nExample: on any finite set $X$ with at least $2$ points, let $R_0$ be the diagonal and $R_1$ the set of pairs of distinct points.\nClauses 1 and 2 hold, and clause 3 holds with $p^0_{11} = \\abs{X} - 1$ and $p^1_{11} = \\abs{X} - 2$, while every other $p^k_{ij}$ is $0$ or $1$.\n\nNon-example: let $X = \\set{a, b, c}$ with $R_0$ the diagonal, $R_1$ the edges of the path $a - b - c$ in both directions, and $R_2 = \\set{(a, c), (c, a)}$.\nClause 3 fails for $i = j = 1$, $k = 0$: for $(a, a)$ there is one $z$ with $(a, z), (z, a) \\in R_1$, but for $(b, b)$ there are two.\n\nA symmetric scheme with $2$ classes is the same as a [[strongly regular graph|C22WGRP3T8]] that is neither complete nor edgeless, with $R_1$ as its edge relation.\n",
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