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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. For Nakayama algebras with Kupisch series [n,n,n-1,n-2,...,2,1] the sequence starts with 8, 75, 864, 12005 and might be given by https://oeis.org/A071720.

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    $\begingroup$ For $D_6$, one gets 228055 = 5 * 17 * 2683. This does not match the number of maximal chains in the noncrossing partition lattices. $\endgroup$
    – F. C.
    Commented Nov 5, 2020 at 20:26
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    $\begingroup$ For $E_6$, one gets $846720 = 2^7 * 3^3 * 5 * 7^2$. $\endgroup$
    – F. C.
    Commented Nov 5, 2020 at 20:26
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    $\begingroup$ There seems to be some divisibility by h/2, where h is the Coxeter number. The Auslander-Reiten translation acts on the set of bases for the K-group. The order of every orbit divides h. Not clear to me what the stabilizers can be. $\endgroup$
    – F. C.
    Commented Nov 6, 2020 at 10:40
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    $\begingroup$ And for $E_7$, one gets $221714415 = 3^6 * 5 * 13 * 4679$. $\endgroup$
    – F. C.
    Commented Nov 9, 2020 at 7:18

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