Is it obligatory for the son and his brothers to seek their father's permission if the father is divorced and pays a monthly alimony?

The meaning of "seeking permission" is not clear. However, if it refers to seeking permission upon leaving and for all other actions, then this is not obligatory. What is obligatory is to be dutiful to one's parents, obey them in what is good, and accompany them kindly. Part of perfecting this kind companionship is to seek their permission when leaving and before undertaking any significant matter, especially if doing so would please them and increase their satisfaction. An adult, mature son is not under custody. As for young children, their mother's custody does not prevent their father from caring for and raising them.