We revisit the Strong Birthday Problem (SBP) introduced by DasGupta'05, which asks for the minimum population n required such that, with a probability of at least 1/2, every individual in the group shares a birthday with at least one other person. Formally, we develop and analyze computational frameworks to determine the probability that in a group of n people with birthdays distributed over m days, each day either has two or more birthdays or is birthday-free. We derive both counting-based and probability-based recurrence relations to solve this problem and establish a novel connection to associated Stirling numbers of the second kind. This relationship is exploited to derive new, more efficient recurrences. Finally, we implement these recurrences using dynamic programming, provide analysis of their asymptotic complexities, and present numerical evaluations that demonstrate the practical efficiency and scalability of our proposed approaches.

翻译：我们重新审视了由DasGupta'05提出的强生日问题（Strong Birthday Problem, SBP），该问题要求确定最小人口数n，使得在至少1/2的概率下，群体中的每个个体都与至少另一个人共享生日。形式上，我们开发并分析了计算框架，以确定在将生日分布在天数为m的n人群体中，每个日期要么拥有两个或更多个生日，要么没有生日的概率。我们推导了基于计数和基于概率的递推关系来解决该问题，并建立了与第二类关联斯特林数（associated Stirling numbers of the second kind）的新联系。利用这一关系，我们推导出了更高效的新递推式。最后，我们使用动态规划实现了这些递推式，分析了其渐近复杂度，并给出了数值评估，以证明我们提出方法的实际效率和可扩展性。