counter
nth generation; nth person to hold a position
Counter suffix attached to numbers to indicate the generation of a family business, dynasty, or the ordinal holder of a title or position. Often used for kabuki actors, sumo wrestlers, and company presidents.
See also: 代
彼は五代目の社長です。
He is the fifth president (of the company).
あの歌舞伎役者は十二代目だ。
That kabuki actor is the twelfth generation.