Let $A\_n=kQ\_n$ be the path algebra of linear oriented Dynkin graph $Q\_n$ of Dynkin type $\mathcal{A\_n}$ (so $A\_n$ is the unique hereditary Nakayama algebra given by quiver and relations).


The number of tilting $A\_n$-modules equals the Catalan numbers $C\_n$ when $n$ is the number of simple $A\_n$-modules.



> 
> Question: Is there a natural bijection between Dyck paths of length $2n$ to the tilting modules of $A\_n$? 
> 
> 
> 


I would think that this might be noted somewhere in the literature, but I did not find it.



Natural bijection between Dyck paths and tilting modules
This is more of an expansion of Sam's comments, but too long for a comment itself:


As pointed out by Sam in that Theorem 4.2.2.2, tilting modules of the linear type $A\_n$ quiver are in natural correspondence to triangulations of a regular $(n+3)$-gon.


**Short answer:** Any bijection between triangulations and Dyck paths would work, and there are plenty. Everyone is "natural" in some respect and "unnatural" in others.


**Longer answer:** Triangulations of a polygon might be generalized to all finite type using either of the following:


* noncrossing partitions $NC$
* clusters $CL$
* Coxeter-sortable elements $CS$


and there is a uniform type-independent bijection between them.


On the other hand, Dyck paths might be generalized to antichains in the root poset of a finite crystallographic type $AC$.


One now has
$$|NC| = |CL| = |CS| = |AC|$$
and indeed many refinements of this identity hold (such as the one mentioned in Section 4.7).


The closest connection between the two that is known is (caution, self-reference!) the article [A uniform bijection between nonnesting and noncrossing partitions](https://arxiv.org/abs/1101.1277) jointly with Drew Armstrong and Hugh Thomas.


There is much more to say and more references to give, please look into (again caution!) [Cataland: Why the Fuss?](https://arxiv.org/abs/1503.00710) for all needed notions and references.


