[seqfan] Re: Breaking news on partition numbers.

[peter.luschny]

Omar E. Pol  wrote:
OEP> The model looks like a three-dimensional tree where each
OEP> branch is the head of the outer shell of the partitions of n.

I find this an attractive way to look at the partitions.
Though I certainly would like to understand your trees also
in terms of a poset (or lattice) and see how they can be generated
algorithmically. This seems to be a nontrivial task, however.

OEP> Question: Is there some connection with fractals?

As you can and probably have read in the nice article by
Carol Clark [1] Ken Ono said: "We prove that partition numbers
are ‘fractal’ for every prime. These numbers, in a way we make
precise, are self-similar in a shocking way."

Whether this kind of being ‘fractal’ is related to what you might
see as a ‘fractal’ structure in your pictures I cannot judge.

In any case this ‘fractal’ nature seems to be the key to
understand not only the arithmetical properties of the partition
numbers but also the new formula of Jan Bruinier and Ken Ono.
It enables a computation which is located in an infinite
dimensional space to be 'projected' on a finite dimensional one.
And the trace of this projection is the new formula.

Cheers, Peter

[1] http://esciencecommons.blogspot.com/