We know that Integer Partitions form a poset (actually we can define more than one partial orders on it), and so we can have some kind of Mobius function on it and consequently Mobius Inversion Formula (correct me if I am wrong)..... I am looking forward to some examples of applications of such kind of formula if it exists, and whether we can look Hook length formula as a manifestation of it or not.... Either you can give an example or point out some literature to look into....