In many old families, one character in each person’s name marks the generation. Since the early Ming, the descendants of Confucius have taken these characters from lists issued by the central government. Chinese Wikipedia gives the list for the 56th to the 105th generation, fifty characters in all. It begins 希言公彦承 and ends 永锡世绪昌.
Under the Qianlong Emperor, the court decreed that descendants of Confucius and of three other sage families must take their names from the granted characters, or they would not be entered in the family register.
The Ming founder set twenty characters for each line of his own descendants. According to Chinese Wikipedia, the former premier Zhu Rongji belongs to the generation 镕 in the line of the Prince of Min.