In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by 2×1040. 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 4×1037. 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.

中文翻译：

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没有提升的国际象棋图数的上限

2015 年，Steinerberger 表明，没有提升的合法国际象棋图的数量自上而下限制为 2×1040。这个数字是通过限制主教和棋子的位置以及在没有棋子被捕获时通过精确的界限获得的。我们改进了这个估计，并表明法律图表的数量少于 4×1037。为了实现这一点，我们在一组图表上定义了一个图形和一个棋子排列类的概念，从而产生了一种用棋盘上任意数量的人来限制棋子位置的方法。