The Matching Distributions: Poisson Limiting Forms and Derived Methods of Approximation

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Abstract
SUMMARY: (1) The Matching Problem is reviewed. (2) Stevens’s approach is generalized to yield the explicit results so far known. (3) A continuous range of Poisson limits is derived for the various matching distributions by an elementary method (alternative to Kaplansky’s symbolic proof). (4) The approximations afforded by some of these are improved by orthogonal polynomial fitting. (5) Further Poisson limits are derived for certain general classes of multiple matching distributions; the asymptotic form for the number of Latin Rectangles is derived as a corollary and some numerical comparisons made. (6) Levene’s matching problem is generalized: A general Poisson limit is obtained and some exact distributions derived. Kullback’s matching distribution is generalized and a general Poisson limit obtained.

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