Dividing up the unrestricted partitions

Importance: Medium ✭✭
Author(s): David S.
Subject: Combinatorics
Recomm. for undergrads: no
Posted by: DavidSNewman
on: May 11th, 2010

Begin with the generating function for unrestricted partitions:


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.


Andrews, George E., The Theory of Partitions, Cambridge University Press (1984)

* indicates original appearance(s) of problem.

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?

pentagonal number theorem

Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:


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?

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