The exponential generating function of a sequence $a[n]$ is:
$\displaystyle \text{EG}(a;x) = \sum_{n=0}^\infty a[n] \frac{x^n}{n!}$
My question is about exponential generating functions of binary sequences $b[n]$, i.e. for which the codomain is $\{0,1\}$.
We know for all such $b[n]$, the exponential generating function $\text{EG}(b;x)$ is absolutely monotonic, and furthermore that it is an entire function with no poles, since the function $\exp(x)$ is the special case where $b[n] = 1$ for all $n$. We can compare any two such $\text{EG}(b_1;x)$ and $\text{EG}(b_2;x)$ by saying that $\text{EG}(b_1;x) \leq \text{EG}(b_2;x)$ iff there exists some real $r$ such that the inequality holds for all real $x > r$.
As a result, given two binary sequences $b_1[n]$ and $b_2[n]$, we can say that $b_1[n] \precsim b_2[n]$ iff $\text{EG}(b_1;x) \leq \text{EG}(b_2;x)$.
My questions:
- Is $\precsim$ a total order?
- If so, what is the order type of $\precsim$?
- There seems to be an initial segment of $\Bbb N$, given by the set of all binary sequences with finitely many $1$'s, ordered lexicographically. After this, is it dense?