Maximal green sequences (MGS’s) are combinatorial objects that involve local transformations of directed graphs, also called quivers. We studied minimal length MGS’s for quivers of type A. It is know that such quivers are in bijection with triangulations of polygons. Moreover, these local transformations of quivers behave well under this correspondence, and there is a related notion of mutation of triangulations. This enabled us to study MGS’s both in terms of quivers and in terms of triangulations. We showed that any minimal length MGS has length n+t, where n is the number of vertices in the quiver and t is the number of 3-cycles. We also developed an algorithm that yields such sequences of minimal length.
Journal of Algebraic Combinatorics volume 44, pages 905–930 (2016)