Within and Between Allelic Class Diversity
The term allelic class is defined in Innan and Tajima (1997)


  Suppose that there are two nucleotides, say A and T, in a particular site.Then, we can divideDNA sequences into two classes: one class includes sequences with A and the other includes sequences with T in this site. We call such a class an allelic class


Two elements that were misleading (for me at least) in the article


Implicitly, in the paper of interest (Charlesworth et al. 1997), they define allelic classes for a site that is known to be under selection (and that can only be bi-allelic). The concept of allelic class is not for a polymorphic neutral site they look at but always for the same site (the one under selection).
The formula has an extra parenthesis. The real formula should be $2 \sum x_t(1-x_t) + y_t(1-y_t)$.


What is $x$ and $y$?

Consider a polymorphic (bi-allelic) locus (that is not under selection) and calculate the frequency of one allele in the first allelic class (defined for the site under selection) (this is $x$) and calculate the frequency of the same allele in the other allelic class this is $y$. In other words, consider the selected has two alleles $A$ and $B$ and the neutral locus has two alleles $a$ and $b$. $x$ is the frequency of sequences containing $A$ that also contain $a$ and $y$ is the frequency of sequences containing $B$ that also contains $a$.

Meaning of the formula

The formula is just the sum of the expected heterozygosity within each allelic class. $$2x(1-x) + 2y(1-y) = 2 (x(1-x)+y(1-y))$$. Then, we just have to sum over all time step during which the neutral locus is polymorphic and we get $$\pi_A = \sum_{i=1}^T 2 (x_t(1-x_t) + y_t(1-y_t)) = 2 \sum x_t(1-x_t) + y_t(1-y_t)$$, where $x$ and $y$ are now function of the instant (time) considered and $T$ is the total number of time steps.

What values can $\pi_A$ take?

This measure takes values between 0 and $T$ as all $x_t$ and $y_t$ take values between $0$ and $1$ (and therefore $x(1-x)$ and $y(1-y)$ take values between 0 and 0.5). Dividing the whole thing by $T$ will result in a measure that is bound between 0 and 1.



I would have not been able to understand without the help of @AMR. Thank you