From recursions to asymptotics: Durfee and dilogarithmic deductions

We consider two dimensional arrays p(n,k) which count a family of partitions of n by a second parameter k, usually the number of parts. Such arrays frequently satisfy a finite recursion of a certain form, detailed in formula (2), as well as an asymptotic relation

(∗)

For such situations, we can characterize (Theorem 1) the function g(u) in terms of a polynomial associated with the recursion. We also identify (Theorem 2) a class of families which satisfy both the desired recursion and the limit law (*). For such families, the function g(u) is characterized by Theorem 1, and this resolves a number of conjectures made in an earlier work Electron. J. Combin. 5 (1998) R32] concerning asymptotic enumeration of partitions by the size of their Durfee square. Finally, we study a family of partitions introduced by Andrews Amer. J. Math. 91 (1969) 18–24]. These partitions do satisfy the desired recursion, but it is not known for sure whether they also satisfy the accompanying limit law. We prove (Theorem 3), conditionally on the conjectured limit law holding, some identities involving the dilogarithm. These identities are seen empirically, by calculation to many decimal places, to be true.