Let $R_{n,m}^q$ be the finite dimensional algebra $K\langle x_1,...,x_n\rangle/J^m$, where the field $K$ has $q$ elements and $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring with the ideal $J$ generated by $x_1,...,x_n$.
Let $T_{n,m}^q$ be the finite dimensional algebra $K[x_1,...,x_n]/J^m$, where the field $K$ has $q$ elements and $K[x_1,...,x_n]$ is the commutative polynomial ring with the ideal $J$ generated by $x_1,...,x_n$.
Question: What are the number of non-zero ideals of $R_{n,m}^q$ and $T_{n,m}^q$ for $n,m \geq 2$?
$R_{n,m}^q$ is commutative exactly when $m=2$ (and then coincides with $T_{n,m}^q$) and in this case the number of non-zero ideals seems to be given by the sequence $a_t^q=\sum\limits_{k=0}^{t}{[t,k]_q}$, where $[n,k]_q$ is the Gaussian binomial coefficient (which counts the number of k-subspaces in a vector space of dimension n over a finite field with q elements, which makes this plausible for $m=2$).
I think I asked a similar question before, but this time we are interested in the total number of ideals and not the ideals up to isomorphism.
