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I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ which enumerates the number of integer partitions of $n$: does anybody have the reference of this paper?

And if not a closed-form exactly, then I seem to recall some significant advance was made recently: can you provide any bibliography (2010, 2011)?

Hopefully this will ring a bell with someone...

Thanks!

amWhy
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Jérémie
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3 Answers3

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Looks like you're looking for Bruinier and Ono's recent paper Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. It received a lot of publicity recently.

Qiaochu Yuan
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  • Thanks! That's it!! (And the link to the publicity was helpful in determining this was what I was looking for :-) – Jérémie Jun 27 '11 at 23:11
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There is a rather unkown expression obtained by G.Iommi Amunátegui: A non-recursive expression for the number of irreducible representations of the symmetric group $S_n$", Physica 114A (1982), 361-364, which gives $p(n)$.

alexwlchan
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user84744
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  • The paper is available here: http://arxiv.org/abs/1307.2098. But as far as I can tell, there is no proof, just a conjecture based on some tabulated values... – Hans Lundmark Nov 10 '15 at 07:54
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I doubt you will get an "elementary" closed form but JM Ain't A Mathematician was able to find a sequence of different continuations of the partition function whose limit looks increasingly well behaved. See here

Computing the limit as his $\rho \rightarrow 1$ and then computing a taylor series of the resultant function seems like an attack for producing a non recursive formula for evaluating partitions.

Sidharth Ghoshal
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