Leaps in the depth of compositions of irreducible morphisms

In this article, we give a family of examples of algebras, showing that for every n ≥ 2 n ≥ 2 and m ≥ 0 m ≥ 0 , there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the $$(n+m+3)$$ ( n + m + 3 ) -th power of the radical, but not in the $$(n+m+4)$$ ( n + m + 4 ) -th power. Such an algebra may be also supposed to be string and representation-finite.