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Dividing up the unrestricted partitions
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
I've been thinking about this problem since about 1970. Emory Starke thought that it was a good problem, but not suitable for the Problems section of the AMM, because it was unsolved. George Andrews and Freeman Dyson also thought that it is a good problem, but neither had any ideas how to solve it.
I've found choices of sign which yield series with coefficients 1, -1, or 0 for all exponents about as high as 110 using computer searches. One thing which mitigates against finding a meaningful solution is that there is no known pattern for the number of unrestricted partitions modulo 2.
Bibliography
Andrews, George E., The Theory of Partitions, Cambridge University Press (1984)
* indicates original appearance(s) of problem.
pentagonal number theorem
On May 18th, 2010 Benjamin Young says:
Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:
(1+x)(1+x^2)(1+x^3)...
If you change *all* of the + signs in the above into minus signs, then the statement of your conjecture holds; indeed there is an explicit formula for the terms of the generating function involving the pentagonal numbers, hence the name of the theorem. This theorem has several pretty and well-publicized proofs (see Chapter 1 of the introduction to "The Theory of Partitions" by George Andrews, or Chapter 14.5 of "Introduction to Analytic Number Theory" by Tom Apostol, or "Proofs from the book" by Aigner-Ziegler, or Wikipedia).
I would wager that this observation isn't terribly helpful, but still. Was this was the motivation of the problem?
Drupal
CSI of Charles University
a question on the numerical work
I'm also curious to know a little more about the experimental work that was done -- roughly how many ways were there to choose the signs to make things work up to degree 110? were there any choices which gave you lots of zeros?