This question emerged during an activity I ran for some middle school students a couple weeks ago; basically, it's about a way to "count" - with an appropriate kind of weight - the Egyptian fraction representations of a given rational.
Let $[\mathbb{N}]^{\mathit{fin}}$ denote the set of finite sets of natural numbers. Given a positive rational $q$, let $\mathsf{E}(q)=\{X\in[\mathbb{N}]^{\mathit{fin}}: q=\sum_{x\in X}{1\over x}\}$ be the set of Egyptian fraction representations (without duplication!) of $q$. For example, $\mathsf{E}(1)$ contains both $\{1\}$ and $\{2,3,6\}$, as well as infinitely many other finite sets.
I'm interested in the function $$\mathscr{E}: q\mapsto \sum_{X\in \mathsf{E}(q)} \left( \prod_{x\in X}{1\over x}\right).$$ Intuitively, I'm trying to count the Egyptian fraction representations of $q$; of course this is infinite regardless of what $q$ is (since ${1\over n}={1\over n+1}+{1\over n^2+n}$), but I'm tweaking things to pay more attention to "short" representations with "small" denominators.
In general, $\mathscr{E}$'s behavior seems rather complicated. I have an outline of an argument that $\mathscr{E}(q)$ is indeed finite for all $q$, but I'm not entirely certain it's correct; even if it is correct, anything more than that is unclear to me, including its behavior on specific values. So my two questions are:
Question 1: What is $\mathscr{E}(1)$?
Of course we can trivially work out some lower bounds, e.g. $\mathscr{E}(1)>{37\over 36}$, but I don't see a way to approach a good exact understanding.
Question 2: Is $\mathscr{E}(q)$ in fact finite for all $q\in\mathbb{Q}_{>0}$?
In general, I'm interested in any information about $\mathscr{E}$, or indeed other ways to "count" Egyptian fraction representations. Most of the research I can find is about determining individual Egyptian fraction representations which are "optimal" in some sense (e.g. minimal length), which isn't exactly what I'm looking for. (I can't even find any information about the function sending $(q,n)$ to the number of Egyptian fraction representations of $q$ of length $n$.)