Do the children and their paternal aunts inherit from the deceased who has no children, knowing that the paternal aunts had waived their share in a previous inheritance of his?

If the paternal aunts, being adult and sane, voluntarily relinquish their share of inheritance, their relinquishment is valid and considered a gift. The portion of land that has not been divided, if relinquished by the paternal aunts, becomes part of the share of the father and the paternal uncle, with each owning half. The paternal uncle's share from this portion and from the rest of the land is considered part of his estate and is divided as follows:

Wife: One-quarter (due to the absence of children). Sisters: Two-thirds, divided equally among them. The remainder: Goes to the 'Asabah (agnatic heirs), who are the male children (you and your brothers) to the exclusion of females.

If one of the paternal aunts dies thereafter, her share goes to her heirs. You and your male brothers inherit what remains after the share of the paternal aunts and the paternal uncle's wife, as the Prophet (peace be upon him) said: "Give the prescribed shares to those entitled to them; then whatever remains is for the closest male relative."

One who relinquishes his right to inheritance is treated like a donor. If a person gifts something and then the donee dies, and the donor is among the donee's heirs, then he inherits from him. This is not considered a retraction of the gift, because he acquires ownership of the thing through another means, which is inheritance. This is supported by the Hadith of Buraydah (may Allah be pleased with him): "I gave a slave girl in charity to my mother, and she died." The Prophet said: "Your reward is due, and the inheritance has returned her to you." Ibn Battal transmitted the consensus of scholars that whoever gives a charity and then inherits it, it is lawful for him. Some scholars mention a difference of opinion on the matter, but it is weak.