A Local Limit Theorem in the Theory of Overpartitions

Abstract

An overpartition of an integer n is a partition where the last occurrence of a part can be overlined. We study the weight of the overlined parts of an overpartition counted with or without their multiplicities. This is a continuation of a work by Corteel and Hitczenko where it was shown that the expected weight of the overlined parts is asymptotic to n/3 as n → ∞ and that the expected weight of the overlined parts counted with multiplicity is n/2. Here we refine these results. We first compute the asymptotics of the variance of the weight of the overlined parts counted with multiplicity. We then asymptotically evaluate the probability that the weight of the overlined parts is n/3 ± k for k = o(n) and the probability that the weight of the overlined parts counted with multiplicity is n/2 ± k for k = o(n). The first computation is straightforward and uses known asymptotics of partitions. The second one is more involved and requires a sieve argument and the application of the saddle-point method. From that we can directly evaluate the probability that two random partitions of n do not share a part.

Distinct partitions and overpartitions

“…Overpartitions were introduced by S. Corteel and J. Lovejoy in [8] and have been the subject of many recent studies including G. E. Andrews [2], K. Bringmann and J. Lovejoy [5], W. Y. C. Chen and J. J. Y. Zhao [7], S. Corteel and P. Hitczenko [9], S. Corteel, W. M. Y. Goh and P. Hitczenko [10], S. Corteel and O. Mallet [11], A. M. Fu and A. Lascoux [12], M. D. Hirschhorn and J. A.…”

On a nonlinear relation for computing the overpartition function

“…Overpartitions were introduced by Corteel and Lovejoy in [6] and have been the subject of many recent studies including Andrews [2], Bringmann and Lovejoy [4], Chen and Zhao [5], Corteel and Lovejoy [6], Corteel and Hitczenko [7], Corteel, Goh and Hitczenko [8], Corteel and Mallet [9], Fu and Lascoux [11], Hirschhorn and Sellers [13,14], Kim [16], Lovejoy [18,19,20,21,22,23], Mahlburg [24], Merca [26] and Sills [28].…”

Some Arithmetic Properties of Overpartition k-Tuples

“…As the topic of overpartitions has already been examined rather thoroughly [3,4,5,6,7,8,10,11], we look to new constructions. One such construction is that of an overpartition pair of a positive integer n, defined by Lovejoy [9] as a pair of overpartitions wherein the sum of all listed parts is n. For example, the overpartition pairs of 2 are Lovejoy denoted the number of overpartition pairs of a positive integer n by pp(n), with pp(0) = 1 by definition.…”