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 ...
One way to do it is think about what the smallest computer encoding of a chess position (in bits). The maximum size for a given encoding provides an upper-bound on the number of chess positions. For some really tight upper bounds check out https://codegolf.stackexchange.com/questions/19397/smallest-chess-board-compression/19446#19446. This gives an upper ...
More general answers:
OEIS A172532: Number of ways to place 5 nonattacking knights on an n X n toroidal board.
my book Non-attacking chess pieces (6ed, 2013), page 308, chapter "k Knights on an n x n toroidal chessboard".
Case 1: Both kings are away from the passed pawn: IF Black can get into the square, it is a draw.
Case 2: The Black King is in front of the pawn: depends on the position of the White King
Case 3: Black's king is outside the square, but White has a rook-pawn - If Black's king is in front of the pawn, it is a stalemate, and in some other cases, White's king ...