We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for $n\ge k\ge 3$, the probability that a k-gon cannot be formed from a stick broken into n parts is given by n! over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.

我们提出了一种离散方法来解决从断棒形成多边形的问题，这类似于计算具有某些丢番图不等式的整数边长的多边形。即，我们使用 MacMahon 的分区分析来获得受这些不等式影响的折断棒段集大小的生成函数。特别地，我们使用这种方法来证明对于$n\ge 克\ge 3$，将一根棍子分成n份不能形成k边形的概率由n给出！在广义斐波那契数的部分和的线性组合的乘积上，这个问题在过去被证明很难概括。