Let $A$ be a representation-finite quiver algebra.

When $A$ has $n$ simple modules a basic module $M$ with $n$ indecomposable summands $M_i$ is called a K-generator when the $M_i$ generate $K_0(A)$, that is the dimension vectors of the $M_i$ are linear independent over $mathbb{Z}$. Call the number of K-generators the K-number of $A$.

For example when $A=kQ$ with $Q$ of Dynkin type $A_n$ this is the number of $n$ linear independent $n$-vectors over $mathbb{Z}$ with entries 0 or 1 so that the ones appear in one block in the vectors. Their number should be given by $(n+1)^{n-1}$ which is also the number of parking function.

Question 1: How many $K$-generators are there for the other Dynkin types?

Note that the answer does not depend on the orientation.

For example for $D_4$ we obtain the number 315 and for $D_5$ it is 7712. Can one expect that this is some sort type $D_n$-parking function numbers (if they exist)? Types $E_n$ can be done by the computer but it would be interesting to see a direct proof.

Question 2: What is the K-number for linear Nakayama algebras (corresponding to Dyck paths)?

For example for the Nakayama algebra with Kupisch series (2,2,…,2,1) the even Fibonacci numbers appear.