N.Y. Alt. County Gov't Law § 52
A county which shall adopt the county director form of government shall have a county executive to be called "county director. " The board of supervisors shall appoint the county director. The term of office of the county director first appointed shall be four years from and including the first day of January of the year in which the county director form becomes effective in the county. Successors shall be appointed for terms of four years each. Vacancies occurring otherwise than by expiration of term shall be filled by appointment by the board of supervisors for the unexpired portion of the term. The county director may, when authorized by local law, serve as the head of one or more departments not administered by an elective official but without additional compensation. The foregoing provisions of this section, together with other applicable provisions of this chapter, including those provisions which may be made applicable, shall constitute the structure of the county director form of government authorized to be adopted by this chapter.
N.Y. Alt. County Gov't Law § 52