Zavadskij modules over cluster-tilted algebras of type $\mathbb{A}$

Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $A$. In this setting, we prove that the indecomposable uniserial $A$-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the $n+3$-gon associated with the cluster-tilted algebra of type ${\mathbb{A}}_n$ to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type ${\mathbb{A}}_n$. In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules.