In the paper [Hernandez and Leclerc - Cluster algebras and quantum affine algebras](http://arxiv.org/abs/0903.1452), Section 13.5, it is said that when $\mathfrak{g}$ is of type $A\_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}\_2$ for $U\_q(\widehat{\mathfrak{g}})$ is of finite cluster type $D\_4$. There are 16 cluster variables.


It is said that (1) when $\mathfrak{g}$ is of type $A\_2$ and $\ell=3$, (2) when $\mathfrak{g}$ is of type $A\_2$ and $\ell=4$, (3) when $\mathfrak{g}$ is of type $A\_3$ and $\ell=2$, and (4) when $\mathfrak{g}$ is of type $A\_4$ and $\ell=2$, the corresponding cluster algebras are also finite type cluster algebras. How many cluster variables do we have in the cases (1), (2), (3), and (4)? Thank you very much.



Number of cluster variables
For $A\_k$ of level $\ell$, the cluster types are given by square grids of size $k \times \ell$.


Therefore the types you ask for are $E\_6$ and $E\_8$ (see [Scott - Grassmannians and cluster algebras](https://doi.org/10.1112/S0024611505015571)), for which you can easily find the
number of clusters using the usual formula expressing them in
terms of the exponents (see the Y-system article [Fomin and Zelevinsky - $Y$-systems and generalized associahedra](https://doi.org/10.4007/annals.2003.158.977)
in the Annals for example). The numbers of clusters are 833 and 25080.


The number of cluster variables are smaller: 42 and 128.


