# co.combinatorics – Number of K-generators of an algebra and type \$D_n\$-parking functions

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.