# Number of cluster variables

In the paper Hernandez and Leclerc - Cluster algebras and quantum affine algebras, 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.

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), 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 in the Annals for example). The numbers of clusters are 833 and 25080.