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 + 1, 1 + 1 + 1 + 1) , etc..
A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n.
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.
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