Are there dictionaries in math?
Solution 1:
A dictionary is just a function $\mathrm{Dict}\colon \mathrm{Keys} \rightarrow \mathrm{Values}\cup\{\epsilon\}$ where $\epsilon$ is a "null character" with the understanding that $\epsilon\notin\mathrm{Values}$.
For example, let $\mathrm{Keys}=\{A,B,C,...,Z\}$, and $\mathrm{Values}=\mathbb{Z}$. Then, in your case,
$$ \mathrm{Dict}(x)=\begin{cases} 1 & \text{if }x=A\\ 2 & \text{if }x=B\\ 3 & \text{if }x=C\\ \epsilon & \text{otherwise} \end{cases} $$
Solution 2:
A standard Python dict corresponds to a function $f\colon K\to U$ in mathematics, where $U$ is the set of all possible Python objects or values of built-in types, and $K$ is a finite subset of $U$. I.e., for some finite set $K$ of keys it associates to each key an arbitrary (Python) value. When trying to evaluate a dict on a value that is not in $K$, a KeyError is raised. This corresponds to the fact that in mathematics you can't ask for the value of $f$ on something that is not a member of $K$.
(Note that in Python, None is a first class object/value unlike null in C or Java. Maybe C++ programmers can think of it as an actual immutable object that is actually located at the memory location specified by the pointer NULL=0. So null is the address of None. I just mention this because the accepted answer by par seems to be confused about this point. It doesn't become clear whether $\epsilon$ is supposed to model None or KeyError. Either way something is wrong or at least confusing.)
For the particular example dictionary D, we can define $f\colon \{A,B,C\} \to U$ by stipulating $f(A)=1$, $f(B)=2$ and $f(C)=3$. The notation AAAA doesn't seem very useful to me except as a shortcut in very specific use cases. In mathematics you would write this as $f(A) + f(A) + f(A) + f(A)$. Of course this only makes sense if $f(A)$ is something for which $+$ is defined.
Solution 3:
Adding to @par 's excellent answer to deal with the second part of the question:
I would like to use the structure for taking sums: e.g. 'AAAA' is interpreted as 1+1+1+1 etc. What kind of notation should I use?
In your case the Keys are characters which you want to think of as Strings that happen to have length 1. The Strings have a natural associative operation - concatenation. (In many languages it's even written with a + sign, although it's not commutative.) The integers come equipped with an associative + too. Then what you want is the assertion that Dict respects those operations: Dict(X+Y) should equal Dict(X)+Dict(Y). There is vocabulary in mathematics for that: Dict is a semigroup homomorphism.
That allows you to extend Dict from the set of one character Strings to the semigroup of all finite Strings.
If that is in fact what you plan to do you should not call your function "Dict". Consider something like "Hash" (since it does assign an integer to a string). Bury the dictionary that gives the value of the hash function on single characters privately inside the definition of a Python object Hash. Then compute Hash.value(s) by looping over the characters in s, looking up their values in the dictionary and adding.