The free spectra of varieties generated by idempotent semigroups

We give an exact formula for the logarithm of the free spectra of band varieties. We show that the the logarithm of the size of a free algebra in a band variety is $\frac{4}{(k-3)!}n^{k-3}\log n-\frac{4}{(k-3)!}n^{k-3}\sum_{j=1}^{k-3}\frac{1}{j}+ O(n^{k-4}\log n)$, where $k$ is a constant depending on the variety.