Is it permissible for a man to give up his seat on public transportation for someone else to sit there?

There is no objection to a person standing up from their seat in transportation or elsewhere to let someone else sit there, if it is done willingly. This act is considered altruism (e.a. Ithar) and a commendable good deed in Islamic law, especially if the person being given preference is among the weak or elderly. There is also no objection to someone sitting in the vacated spot if they know the person stood up willingly. However, it would be more virtuous to refrain from sitting in it out of piety, as Ibn Umar (may Allah be pleased with him) used to do.