A photocopy of a sheet of paper in my handwriting. At the top left, my initials are written in the handwriting of Jaap Seidel. The page begins as follows.

**Theorem.** Let *X* be a set of points in

the *n*-cube *Q*_{n} |
the Johnson scheme *J*(*k,n*) |
the symmetric group *S*_{n}. |

Then the average distance between two members of *S* is not greater than

Equality holds if and only if *n* is

an orthogonal array of strength 1 |
a 1-design |
uniformly transitive. |

Then follow three columns giving the proofs of the three propositions (completely elementary, basically Cauchy–Schwarz), followed by some remarks about what conditions a P&Q-polynomial association scheme would have to satisfy for a version of the first two theorems above to hold (basically, the inner product of projections onto the first eigenspace should be a monotone decreasing function of the distance).

There should be more to it, since even such a result would not cover the third case.

And the small mystery: why do I have a photocopy annotated by Jaap? I suspect that I said something like this in conversation at a conference, Jaap was interested, so I wrote out a proof and gave it to him, and he decided firmly that I should have a copy myself. When could this have been? My guess is the Montréal conference on algebraic, extremal and metric combinatorics in the late 1980s, but I am not certain.

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About Peter Cameron

I count all the things that need to be counted.