N.Y. Pub. Off. Law § 9
Every deputy, assistant, or other subordinate officer, whose appointment or election is not otherwise provided for, shall be appointed by his principal officer, board or other body, and the number thereof, if not otherwise prescribed by law, shall be limited in the discretion of the appointing power. If there is but one deputy, he shall, unless otherwise prescribed by law, possess the powers and perform the duties of his principal during the absence or inability to act of his principal, or during a vacancy in his principal's office. If there be two or more deputies of the same officer, such officer may designate, in writing, the order in which the deputies shall act, in case of his absence from the office or his inability to act, or in case of a vacancy in the office, and if he shall fail to make such designation, the deputy longest in office present shall so act. If two or more deputies present shall have held the office for the same period, the senior deputy in age shall so act. Such written designation by a state officer shall be filed in the office of the secretary of state; and by any other officer, in the office of the clerk of the county in which the principal has his office. If a vacancy in a public office shall be caused by the death of the incumbent, the deputies shall, unless otherwise provided by law, continue to hold office until the vacancy shall have been filled in accordance with law.
N.Y. Pub. Off. Law § 9