Elucidation/intuition of atoms/molecules in combinatorics I have in my notes the following passage. For context, the lecture is on the Exponential Generating Function.
Suppose $a_1,a_2...$ is some sequence of numbers. Then the exponential generating function of $(a_n)$ is $E(a_n,t)=\sum_n \frac{a_nt^n}{n!}$
Some examples are;
The sequence $a_n=c^n$  for some $c \in \mathbb{C}$ is generated by $e^{ct}$ [I suppose because you could recover $a_n$ by n (or n-1) fold differentation].
The sequence $-1^{n/2}\chi({n \in 2\mathbb{Z}})$ is generated by $cos(cx)$ [for the same reasons as above?]
Then this mysterious passage appears;
''Suppose $C_n$ is a set which can be associated with a set $S$ with $|S|=n$. Denote $\mathcal{C}=\bigcup_{n}C_n$. Then we denote the elements of $\mathcal{C}$ as ''atoms''. A molecule with $c$ components on a set $S$ is a partition of $S=\bigcup_1^c B_i$ where each $B_i$ is associated to an atom. So a molecule is: $\{(B_1,a_1),(B_2,a_n)...(B_c,a_c) s.t. a_i \in C_{|B_i|}$}
The EGF of $\mathcal{C}$ is defined to be:
$$f(\mathcal{C},t)=\sum_n \frac{|C_n|t^n}{n!}$$
The rest of the lecture is about theorems regarding atoms/molecules (I will gladly share if this is necessary) which is largely indecipherable to me.
My question then is...what? Can someone provide me with a concrete example because I cannot parse this out. The only thing I can think of is that if $C_n$ could be something like the ''the set of words from an alphabet  (so this is $S$?) of size $n$''. So one atom in this case is $C_1=\{a\}$. Another would be $\bigcup_1^3C_n$, all the words of with 3 or less letters? Ok this seems plausible, but that what is a molecule in this case? I'm very unsure/confused.
 A: Here are three examples.

*

*For every $n$, there is exactly one atom of size $n$. Then a molecule on a set $S$ is just a partition of $S$.

*For every $n$, there are $(n-1)!$ atoms, which are the cyclic permutations of $\{1,2,\dots,n\}$. Then a molecule on a set $S$ is a permutation of $S$, written in cycle notation. For example, one possible molecule on the set $\{a,b,c,d,e\}$ has the partition $\{a,c\} \cup \{b,d,e\}$ with associated atoms $(1\;2)$ and $(1\;3\;2)$. This is a convoluted way of encoding the permutation $(a\;c) \; (b\;e\;d)$.

*For every $n$, there are $n^{n-2}$ atoms, which are the spanning trees of the complete graph $K_n$. Then a molecule on a set $S$ is a forest with vertex set $S$. To specify a forest with vertex set $S$, we first specify its connected components (that's the partition $B_1, B_2, \dots, B_c$) and then we give a spanning tree of each component.

In the second and third case, you see a common theme: given a set $S$ of size $n$ and an atom $A \in C_n$, the pair $(S, A)$ represents the atom $A$, but "labeled" by the elements of $S$. For example, if $S = \{b, d, e\}$ and $A$ is the cyclic permutation $(1\;3\;2)$, then $(S,A)$ represents the cyclic permutation $(b\;e\;d)$ of $S$.
In Wilf's classic textbook generatingfunctionology, the terminology used for this idea is "cards", "decks", and "hands", instead. An atom is a card; a molecule is a hand; a set of atoms $\mathcal C$ is a deck.
