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A local limit theorem in the theory of overpartitions
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http://hdl.handle.net/1860/1634

Title:  A local limit theorem in the theory of overpartitions 
Authors:  Corteel, Sylvie Goh, William M.Y. Hitczenko, Pawel 
Keywords:  Partitions;Combinatorial probability;Local limit theorem;Asymptotic analysis 
Issue Date:  2006 
Publisher:  Springer Verlag 
Citation:  Algorithmica, 46(34): pp. 329343. 
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 ! 1
and that the expected weight of the 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. 
URI:  http://www.doi.org/10.1007/s004530060102z http://hdl.handle.net/1860/1634 
Appears in Collections:  Faculty Research and Publications (Mathematics)

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