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A local limit theorem in the theory of overpartitions
Please use this identifier to cite or link to this item:
http://hdl.handle.net/1860/1634
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| 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(3-4): pp. 329-343. |
| 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/s00453-006-0102-z
http://hdl.handle.net/1860/1634 |
| Appears in Collections: | Faculty Research and Publications (Mathematics)
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