Is it permissible for a man married to four women to marry immediately after divorcing one of them? And why?

If the divorce is revocable, then the man is not permitted to marry [another] until the 'iddah (waiting period) of the fourth divorcee has elapsed, by agreement, because she is still considered to be his wife. However, if the divorce is irrevocable, there is a difference of opinion:

The School of Malik and Shafi'i: He is permitted to marry a fifth wife or the sister of the irrevocably divorced woman, because the marital bond and inheritance rights between them have been severed. The School of Abu Hanifa and Ahmad ibn Hanbal: He is not permitted to marry until the 'iddah of the irrevocably divorced woman has elapsed. This view was narrated from Ali, Ibn Abbas, and Zayd ibn Thabit, and was also held by Sa'id ibn al-Musayyib, Mujahid, Al-Nakha'i, Al-Thawri, and the People of Opinion (Ahl al-Ra'y).