1. The Generalized Birthday Problem
2. Probabilities for a Generalized Birthday Problem
3. Note that the principal argument leading to (2) — that La is identically 0, because the set, is composed of ‘pairs’ of realisations having opposite sign (σ)'s - is valid only if the restriction, is imposed for all i. It may be seen, however, that if the restriction is imposed only for certain i, e.g., i = L′ + 1 and L′, there will be ‘pairs’ of realisations in only one of whose members satisfies the restriction. Since the determinant, |e′ij |, considers only realisations satisfying the restriction, these ‘pairs’ would not cancel each other out, as they must if |e′ij | is to count precisely those realisations having no coincidences.
4. Hwang F. K. (1974) A discrete clustering problem. Unpublished memorandum.
5. Defined by first choosing the lowest value of j. and then the corresponding lowest value of i such that xi(j) = x i − 1 (j).