# Generating functions with all non-zero coefficients equal to one

Inspired by this question, I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ has significant applications, as do monomial symmetric functions but for the purposes of this question, we should ignore them. As Graham has commented, indicator functions also fall into this category. The best formulation I can think of for why these should be ignored is that they are most interesting for the purpose of taking products of generating functions, rather than being directly used for computation.

More specifically, such a generating function would (in my mind) have to be multi-variate, enumerate some object of interest and facilitate computations related to that object in a situation where direct computation is not straightforward. As an example of what I would consider cheating, by specializing many variables to one, the generating function in the aforementioned question allows for computing joint distributions of entries in a reduced word with great ease. However, to compute the generating function, it seems to me one would have to enumerate all such words anyways, hence no labor is saved. Were this not the case, this function would be an excellent example.

What would be an example of such a generating function where computation is assisted without being embedded in constructing the generating function? Even if the generating function serves as a useful book keeping device, that would be okay.

Please comment below with any suggested improvements for what should define a "useful" generating function. Explanations for why such a function cannot exist are welcome as well.

Edit 1: Added Graham's comment on indicator functions.

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I was actually thinking of a similar question the other day, although I was approaching it from the viewpoint of whether there is some nice $\mathcal{S}$- (or $\mathcal{A}$-) regular language for which there is a finite complete set of ($\mathbb{N}$-valued) invariants; the values of these invariants would of course be the exponents of the variables in each monomial. On the other hand, I suppose I may be restricting myself too much by asking for a regular language as this necessarily forces the associated generating function to be rational... – ARupinski Jul 23 '13 at 1:23
You probably mean to say that all coefficients are equal to zero or one. In which case, in $n$ variables, you can identify your generating function with the indicator function of (any) subset of lattice points in ${\mathbb N}^n$. So your question is, in this sense, universal. Can you narrow it? – Graham Denham Jul 23 '13 at 2:06