About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster

Question

Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a sequence $m=(x_1,\dots,x_k)$, $k \geq 1$, consisting of distinct exchangeable variables from $ex$ a simple sequence and let $\mu_m:=\mu_{x_k} \circ \cdots \circ \mu_{x_1}$. Is the equality \begin{equation} \mathcal{X}_{\Sigma}=\{y \in \mathcal{X}_{\Sigma}|\mu_m(x)=y \;\text{for some simple sequence}\; m \;\text{and some} \; x \in ex\} \end{equation} true in general? The $\supseteq$ part is trivial, but the other way around confuses me. So for instance, for a type $A$ cluster algebra this should be true, but I am not sure about other types. Any help in a form of a hint/explanation/reference/counter example would be much appreciated.

Answer 1

Not really an answer, but a comment about the related question:

Can one reach every cluster in that way ?

It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.

A conjecture for the number of reached clusters for this initial quiver in type $D_n$ can be found in this OEIS sequence.

EDIT:

Even in type $A_4$, not all clusters are reached if the initial cluster is not the equioriented Dynkin quiver.

Here is a small sage program that generates the set of reached clusters:

def recursive_comb(carquois):
    """
    EXAMPLES::

        sage: Q = ClusterQuiver(DiGraph({0:[1]}))
        sage: len(list(recursive_comb(Q)))
        5
    """
    n = carquois.n()
    seed = ClusterSeed(carquois)
    initial_cluster = seed.cluster()

    def voisins(s):
        c = s.cluster()
        for i in range(n):
            if c[i] in initial_cluster:
                yield s.mutate(i, inplace=False)

    return RecursivelyEnumeratedSet([seed], voisins, structure='graded')

EDIT:

Using the same program, one can see that some cluster variables are missing already for some choices of initial quiver in type $D_4$:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c'],'c':['d','e']}))
sage: Q2 = ClusterQuiver(DiGraph({'c':['b','d','e']}))
sage: len(set(v for c in recursive_comb(Q1) for v in c.cluster()))
16
sage: len(set(v for c in recursive_comb(Q2) for v in c.cluster()))
15

EDIT:

Even in type $A$, some clusters can be out of reach:

sage: Q1 = ClusterQuiver(DiGraph({'b':['c','a'],'c':['d']}))
sage: len(list(recursive_comb(Q1)))
37

instead of 42, the full number of clusters.