I am trying to understand conceptually the differences between ordinary generating functions (OGF$=1+x+x^2+\ldots$ ) and exponential generating functions (EGF$=1+x+\frac{x^2}{2!}+\ldots$ ) when it comes to counting objects (e.g. how labeling and ordering come into play). The easiest way for me to understand the differences is through understanding interpretations of "analogous" representations. Are the following representations correct? Are they good examples for understanding the differences of the generating functions? Are there other good \ better examples?
(a) OGF: Representing an $n$ letter sequence using 1 letter
$$1+x+x^2+\ldots$$
EGF: Representing an $n$ letter word using 1 letter
$$1+x+\frac{x^2}{2!}+\ldots$$
(b) OGF: Representing an $n$ letter sequence where each letter can be one of $k$ possible letters and the order of the letters within the sequence does not matter
$$(1+x+x^2+\ldots)^k$$
EGF: Representing an $n$ letter word where each letter can be one of $k$ possible letters
$$(1+x+\frac{x^2}{2!}+\ldots)^k$$
(c) OGF: Representing a sentence with $m$ words where each word has $n$ letters where each letter can be one of $k$ possible letters and the order of the words in the sentence does not matter and the order of the letters within a word does not matter
$$(1+x+x^2+\ldots)+(1+x+x^2+\ldots)^2+\ldots$$
EGF: Representing a sentence with $m$ words where each word has $n$ letters where each letter can be one of $k$ possible letters (order matters in the sentence and in each word)
$$(1+x+\frac{x^2}{2!}+\ldots)+(1+x+\frac{x^2}{2!}+\ldots)^2+\ldots$$
(d) others?
Best Answer
The best explanation of the differences I've seen are in the discussion of the symbolic method, as given by Flajolet and Segdewick in Analytic Combinatorics, and more accessibly by Sedgewick and Flajolet in "Introduction to the Analysis of Algorithms" (Addison Wesley, 2nd edition 2013). This Wikipedia article should be enough to get you going.