A power series identity

Last year, I was giving a talk for PuMaGraSS, and I wanted to include the following power series identity:

\prod_{n=1}^\infty (1-q^n)=\sum_{n\in\mathbb{Z}}q^{\frac{3n^2+n}{2}}.

The standard way that I have seen this proved is to go via the Jacobi Triple Product Formula which tends to have non-trivial proofs. I decided though that I wanted to find a more combinatorial way of proving this special case. This was successful and can be found at this site, though the proof would be more illuminating with pictures.

Essentially what one does is naively expands the product to get a sum over all paritions with distinct parts and then constructs an involution on a large subset of these partitions to cancel most of the terms, leaving only those that appear on the right hand side of the identity.

I had originally hoped for a combinatorial proof exploiting the fact that the product on the left hand side is the inverse of the generating function for the partition function. Such a proof does exist, is extremely elegant and can be found, amongst other gems in Aigner and Zagier’s book Proofs from the Book. This book is really an amazing read, consisting of a compilation of some of the most elegant pieces of mathematics known to man, and I would highly recommend that anyone with an interest in maths reads it.

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