The first contribution to this issue is An Upper Bound for the Number of Chess Diagrams without Promotion by Daniel Gourion. It shows that the number of legal Chess diagrams is less than 4×1037 which is an improvement on the previous upper bound of 2×1040 by Steinerberger. The trick is to define a graph on the set of diagrams and to classify pawn arrangements.
The estimation of the number of Chess diagrams goes back to the seminal paper of Shannon in 1950. This is related to the work of Tromp on the number of legal Go positions. Tromp also recently investigated the number of legal Chess positions with a randomized algorithm. The code written by Gurion to compute the upper bound is available....