## Question

I have a quick question for anyone familiar with the theory of integer partitions. Because of the geometric series, it is the case that

$\prod\limits_{k \geq 1} \frac{1-x^{3k}}{1-x^k} = \prod\limits_{k \geq 1} (1+x^k+x^{2k})$.

This is a generating series for a certain class of integer partitions, the partitions where no part is divisible by three. By examining the representation on the right-hand side, it seemed clear to me that this was also the generating series for the partitions of $n$ where each even part occurs at most twice, and each odd part occurs at most once.

However, this is apparently wrong, and it is indeed the generating series for the partitions of $n$ simply where each part occurs at most twice. My question is this: why? I can’t possibly conceive of a way you could get a partition where 3 occurs twice. Where any even number occurs twice? Sure. But not any odd one. I mean, the language seems to be

$\{\epsilon,1,2\}\{\epsilon,2,4\}\{\epsilon,3,6\}\ldots\{\epsilon,n,2n\}\ldots$

So what do you think?

just another guy trying to make the diagrams commute.
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### 3 Responses to Question

1. Simme says:

You should consider the {0, 3, 6} term to represent 3 being picked either 0, 1, or 2 times, which might clarify the matter a bit.

2. dx7hymcxnpq says:

Thanks for the post. Is it incorrect to say that it’s the generating series for the partitions of n where each even part occurs at most twice, and each odd part occurs at most once, though? They both seem like valid interpretations of that generating series…

3. dx7hymcxnpq says:

I think the source of my confusion has to do with unique representation in the language. When looking at the generating series (1+x+x^2)(1+x^2+x^4)…, perhaps this is the reason why we should interpret it as representing the partitions of the form
{empty,1,1+1}{empty,2,2+2}{empty,3,3+3}…{empty,n,n+n}…
rather than those of the form
{empty,1,2}{empty,2,4}{empty,3,6}…{empty,n,2n}…?

(In the latter case, we don’t have unique representation of a partition as an element of the language.)