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 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? for all needed notions and references.

Start with a tilting module for the quiver $0\to 1\to 2\to ...\to n$.
Throw away the components with support at 0.
You get a support tilting module $T_1,...,T_k$ for $1\to 2\to ...\to n$.
Take, for each $T_i$, the part $\lambda_i$ equal to the top of $T_i$.
For example, every injective module gives $\lambda_i=1$.
Take the Young diagram of that partition.
The shadow of the Young diagram is the corresponding Dyck path of length $2n+2$.

Conversely, take a Dyck path of length $2n+2$. Then each down-step $D_i$ corresponds to a component $T_i$ of the corresponding tilting module.
The top of $T_i$ is at vertex $k$ where $k$ is the number of upsteps after $D_i$.
If $D_i$ goes from level $\ell+1$ to level $\ell$, the length of $T_i$ is equal to the number of upsteps since the last time the Dyck path went up to level $\ell$ (if none exists, $T_i$ is projective). For example, $T_{n+1}$ will be the projective injective $P_0$.