N.Y. Election Law § 15-106
1. a. In a village where the rules of any party require that any nomination of candidates for village office by such party be made at a primary election, pursuant to this article;
(b) On a Tuesday as soon as practicable:
(iii) Whenever vacancies in the village's board of trustees constitute a majority of the seats thereof, but in no event less than sixty-one nor more than ninety-five days after the day on which such vacancies first constitute such majority. b. In any other village, special village elections shall be held;