N.Y. Const. art. XIII, § 3

The legislature shall provide for filling vacancies in office, and in case of elective officers, no person appointed to fill a vacancy shall hold his or her office by virtue of such appointment longer than the commencement of the political year next succeeding the first annual election after the happening of the vacancy; provided, however, that nothing contained in this article shall prohibit the filling of vacancies on boards of education, including boards of education of community districts in the city school district of the city of New York, by appointment until the next regular school district election, whether or not such appointment shall extend beyond the thirty-first day of December in any year.