Serieshelpmate in 19

Here’s a brilliant problem.

Consider the following chess position.

Black is to make 19 consecutive moves, after which White checkmates Black in one move. Black may not move into check, and may not check White (except possibly on his last move). Black and White are cooperating to achieve the aim of checkmate. (In chess problem parlance, this problem is called a serieshelpmate in 19.) How many different solutions are there?

This problem is due to Kauko Väisänen, and appears in A. Puusa, Queue Problems, Finnish Chess Problem Society, Helsinki, 1992 (Problem 2).

Hint: the above is quoted from Richard Stanley’s Enumerative Combinatorics.