Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points. A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and the dimension vectors of those summands are linear independent (meaning they form a basis of $K_0$). It it known that such modules $M$ are in bijection to parking functions and thus are enumerated by $(n+1)^{n-1}$. ( see A canonical bijection from linear independent vectors to parking functions )
We look at the statistic on $K_0$-basis modules $M$ that associates to $M$ the statistic $dim Ext_B^1(M,M)$.
If we count the number of such $M$ with $dim Ext_B^1(M,M)=0$, we exactly get the tilting modules which are enumerated by the Catalan numbers.
Question: What is the number of $K_0$ basis modules $M$ with $dim Ext_B^1(M,M)=1$?
The sequence starts with 1,6,27,112,450 and might be https://oeis.org/A220101.
Question: More generally, how many $K_0$-basis modules are there with $dim Ext_B^1(M,M)=t$ for a given $t$?
