An upper bound for the number of chess diagrams without promotion

The number of legal chess diagrams without promotion is bounded from above by 2 x 10 40. This number is obtained by restricting both bishops and pawns position and by a precise bound when no chessman has been captured. We improve this estimate and show that the number of diagrams is less than 4 x 10 37. To achieve this, we define a graph on the set of diagrams and a notion of class of pawn structure, leading to a method for bounding pawn positions with any number of men on the board.