In an effort to classify chess games, I thought it would be useful to identify a set of positions by their unique pawn structure that would form a family of positions. There are indeed many positions but my rough calculations indicate 10^12 to be a upper limit of all possible combinations (I may be very wrong as I don't trust my calculations). But interestingly, it is way lower than the number of sensible position,s i.e. ~10^46. I am very interested to know the actual number of pawn structures (mwithout promotion). I guess actual number of legal positions would be difficult to estimate but a general idea might be good.
To classify pawn formations, a good source is chess master books that explore that topic. The book by Boris Persits comes to mind, for example.
I add that it seems impractical as you have been told, to make an exhaustive index of all the combinations that pawn chains can take.
However, it is useful to connect families of openings (classified by ECO, or by open / closed / flank, positional or tactical) and also by the types of endings that can be found according to the pawn structure.
This information has didactic utility for chess plans. There are books that touch on the subject of pawn structures and chess plans.
The pawn chain structures could be described with the characteristics: doubled or solitary / past / advanced / delayed / blocked pawns, V chain or inverted V chain, castling with / without fiancheto and other descriptors that I cannot imagine.
In this way, the number of pawn formations is much less than combinatorial theory can estimate.
For unique pawn structures I would not recommend going by any formula where you will have 10 to the power of n or any indices calculation. The best way to understand pawn structures is to know multiple openings in. There are more than 2000+ opening variations where the middlegame differs in many aspects along with the piece placement. It is always good to understand the patterns of the pawn structures by playing the openings and learning the middlegame.
This way, you can prepare some of the openings and understand the pawn structures which suit your style. This way, GMs prepare themselves. No one has a direct number of exactly how many pawn structures there are. Not even Kasparov or Carlsen might be interested in that unless you are able to show the perfect differences between each kind of pawn structure.