Consider that we use the bubble-sorting algorithm to sort a string of size $n$. We know then that the maximum number of swaps results when the string is in reverse order- this gives $\frac{n(n-1)}{2}$ swaps. There are a total of $n!$ strings. Can we generalize the number of strings that can be sorted in a minimum of any given number of swaps between 0 and $\frac{n(n-1)}{2}$ inclusive? Looking at small cases, I see that this list seems symmetric- for $n=3$, there is 1 string that can be sorted in 0 swaps, 2 strings that can be sorted in a minimum of 1 swaps, 2 that can be sorted in a minimum of 2 swaps, and 1 that can be sorted in a minimum of 3 swaps. So our list for $n=3$ is $1,2,2,1$. Perhaps if we create pyramid of the lists for all $n$ we might be able to find some patterns? The list for any $n$ should sum to $n!$.
Finding patterns/repetitions using the definition of $G$ given here might be productive, perhaps.