Finite formula found for partition numbers

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Credit: Emory and Ken Ono

So this isn’t physics*, but if you squint hard enough, you can probably make a connection.  The hot topic today is Ken Ono‘s latest work on the partition function:

Ken Ono, Amanda Folsom, & Zach Kent (2011). l-adic properties of the partition function American Institute of Mathematics.

Ken Ono & Jan Bruinier (2011). AN ALGEBRAIC FORMULA FOR THE PARTITION FUNCTION American Institute of Mathematics.

A EurekAlert press release appeared today, entitled: New math theories reveal the nature of numbers and people are already whispering “Fields Medal”. Now, I haven’t thoroughly read the paper yet, but, since I’m not a number theorist, my commentary probably won’t change very much anyway.  Obviously, like most press releases, this one is full of hyperbole and ridiculous sentences like, “the team was determined go beyond mere theories”, but the actual work being discussed is fascinating.

Now, when we talk about a partition function in the context of Ono’s work, we don’t mean the partition function that is familiar to most physicists, we mean what number theorists call a partition function.

In this setting, a partition is a way of representing a natural number n as the sum of natural numbers (ie. for n = 3, we have three partitions, 3, 2 + 1, and 1 + 1 + 1, independent of order).  Thus, the partition function, p(n), represents the number of possible partitions of n.  So, p(3) = 3, p(4) = 5 (for n = 4, we have: 4, 3 + 1, 2 + 2, 2 + 1 + 11 + 1 + 1 + 1) , etc..

To be slightly more technical, from Ken Ono and Kathrin Bringman [1],

A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n.

The concept is straight forward, but how to obtain these partition numbers, in general, is actually no trivial matter.

The master of series, Leonhard Euler, worked on solving this problem, to less than fully satisfying results.  Using the reciprocal of what is now called Euler’s function, we get the generator for p(n) by this infinite product,

sum_{n=0}^{infty} p(n)q^n= prod_{n=1}^{infty}frac{1}{1-q^n}.

Here, q^n counts the number of ways to write, n = a_1 + 2a_2 + 3a_3 +ldots, for a_i in mathbb{N}, where each number i appears a_i times.

Obviously, for large n, this can be unwieldy, and it doesn’t lead to an explicit formula, but as long as you didn’t need more than 200~ partition numbers, it was okay.

Mathematics had to wait until the early 1900s before anyone was to expand on Euler’s partition number generator, when Srinivasa Ramanujan made contact with G.H. Hardy.  Ken Ono actually has a beautiful historical, and mathematical, account of the Ramanujan and Hardy story, called “The Last Words of a Genius” [pdf].

Ramanujan famously proved an unusual and surprising result that [2],

p(5n + 4) = 0 (mod 5),
p(7n + 5) = 0 (mod 7),
p(11n + 6) = 0 (mod 11).

He was also responsible for the first attempt at an explicit, although not exact, formula for p(n) with Hardy,

p(n)simfrac{exp(pisqrt{2n/3})}{4nsqrt{3}} as n rightarrow infty.

A decade later, Hans Rademacher came up with an exact formula, involving a convergent series, Dedekind eta functions, and Farey sequences; it was computationally unpleasant, to say the least (and not worth TeXing in here, but see Wikipedia if interested).  It was also not substantially more useful than Euler’s initial work (although more direct).

In 2007, Ono was an author of a paper [1] that provided an arithmetic reformulation of Rademacher’s formula, using a Maass–Poincaré series.  Based on some discussion, it wasn’t a giant improvement over Rademacher work.

It seems that since Euler initially came up with his generating function, there haven’t been major leaps in our understanding of partition numbers.

Apparently that all changes tomorrow.  Ken Ono and colleagues, Jan Bruinier, Amanda Folsom and Zach Kent, will be announcing results that include a finite, algebraic formula for partition numbers thanks to the discovering that partitions are fractal. Well, so what does this mean, for partition numbers to be fractal?

Ken Ono, in the press release:

The sequences are all eventually periodic, and they repeat themselves over and over at precise intervals.

Alright, so obviously without going deeply into paper we can’t go further here (check out the pdf), but one can see how this insight could make the generation of partitions simple and explicit.  This also, apparently, explains, and is linked to, Ramanujan’s congruences above.  How? Well, they’re part of this pattern.

Ken Ono, in the press release:

I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It’s the finite, algebraic formula that we have all been looking for.

Cool.

There is already an extension on the Ono-Folsom-Kent fractal issue by John Webb called, “An improved “zoom rate” for the Folsom-Kent-Ono l-adic fractal behavior of partition values” [pdf here].

*The physics tie in? Alright, so this is reaching, but here we go.  Partitions are visualized using Young tableaus, and anyone in particle physics (see pdf for relevant introduction) has probably come across this, as well as other forms of group representation theory.  Could an ability to always explicitly write down partition numbers translate to physics? I couldn’t ever imagine using groups so large that this could at all come into play… but, one could possibly draw some conclusions about the fractal nature of… ah, I give up.

Update: Comments on connections to actual physics can be found here.

Related Literature:

[1] KATHRIN BRINGMANN, & KEN ONO (2007). An arithmetic formula for the partition function Proceedings of the American Mathematical Society, 135, 3507-3514

[2]Ken Ono (2010). The Last Words of a Genius Notices of the American Mathematical Society, 57, 1410-1419

[3] Folsom A, & Ono K (2008). The spt-function of Andrews. Proceedings of the National Academy of Sciences of the United States of America, 105 (51), 20152-6 PMID: 19091951

[4]Ken Ono, & Jan H. Bruinier (2009). Identities and congruences for the coefficients of Ramanujan’s omega(q) Ramanujan Journal

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