Partition Asymptotics from Recursion Equations

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Abstract
A new method is presented for obtaining the asymptotic behavior, for both n and s large, of the number of partitions of an integer n into s parts of various kinds. It involves solving a recursion equation satisfied by the number of partitions, using asymptotic methods of applied mathematics such as the WKB method, the ray method, and the method of matched asymptotic expansions. It is applied to partitions of n into s parts, into s distinct parts, into srth powers, and into s parts which differ by at least d. The method can be applied to many other problems in partition theory and in combinatorics.

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