TITLE: Natural bijection between Dyck paths and tilting modules
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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 $P_0$.