Distribution of integer partitions with large number of summands

The factor c−1 √ n log n is nothing but the asymptotic value of the mean number of summands in a random partition of n, each partition of n being equally likely; cf. [17]. Note that the number of partitions of n with m 1’s is given by p(n −m) − p(n −m − 1) for 0 ≤ m ≤ n; and, consequently, p(n−m) is nothing but the number of partitions of n which have ≥ m 1’s. Thus the factor one plays an important rôle in the counting function p(n,m) when m is larger than the mean value. Our aim in this paper is to show that such a phenomenon holds for more general partitions. The proof of the theorem is rather simple and relies on a method that we previously employed in [10] (for Dirichlet series) for an analytic proof of a result of Nicolas [15]. From the formula (2), it is obvious that our method is based on explicitly isolating the contribution of the factor 1 to the counting function p(n,m). This method is quite general and can be applied to other partition problems; it can be regarded as analytic version of the sieve of Eratosthenes; cf. [14, Ch. IV]. A general theorem under a scheme due to Meinardus [13] will be derived in § 4, which applies, in particular, to partitions into powers. Finally, we further refine the analysis to establish the limiting distribution (with convergence rate) for the number of summands in partitions into parts ≥ k, 1 ≤ k = o ( n1/4(log n)−1 ) , thus extending and improving the results by Auluck et al. [2]. Notation. We shall use the notation [zn]f(z) to represent the coefficient of zn in the Taylor expansion of f(z). The notation [umzn]g(u, z) is then defined as [um] ([zn]g(u, z)). All limits (including O, , o and ∼), whenever unspecified, will be taken to be n→ +∞. The constant c always denotes √ 2/3π.