An upper bound for the number of chess diagrams without promotion

In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by $2\times {10}^{40}$. This number was 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 legal diagrams is less than $4\times {10}^{37}$. To achieve this, we define a graph on the set of diagrams and a notion of class of pawn arrangements, leading to a method for bounding pawn positions with any number of men on the board.