Remarks on lengths of maximal green sequences for quivers of type $\tilde{A}_{n,1}$

Abstract: A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Br\"ustle, G. Dupont and M. P\'erotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. They also considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type $A$ or of type $\tilde{A}_{n,1}$ by using theory of tilting mutation.