Let $b(r,s,q)$ denote the number of two-sided ideals in $J^2/J^s \subseteq F_q \langle x_1,...,x_r\rangle/J^s$, where $A=F_q \langle x_1,...,x_r\rangle$ is the noncommutative polynomial ring in r variables over the finite field with q elements and $J=\langle x_1,...,x_r\rangle$ is the ideal generated by $x_1, ... , x_r$. (Motivation: one obtains exactly the admissible ideals of the quiver with one point and r loops by this) Im searching for some interesting upper $c(r,s,q)$ and lower $a(r,s,q)$ bounds for $b(r,s,q)$, for example to see what might be realistic to obtain with a computer. So the goal is to give interesting inequalities of the form $a(r,s,q) \leq b(r,s,q) \leq c(r,s,q)$. You are also allowed to choose special values that might be interesting like r=2 and q=2 or 3 it that helps. Might be fun if each post tries to improve the bounds of the previous post.

Let me start: $a(r,s,q)=r^2+r^3+...+r^{s-1}$,by looking at the principal ideals generated by one term polynomials and $c(r,s,q)=2^{q^{r^2+r^3+...+r^{s-1}}}$, by looking at all possible subsets of the vector space $J^2/J^s$.

If you want you can also look at asymptotics or the commutative polynomial ring first.

In Algorithm for finding quiver algebras r-algebras the question was about precise numbers. In this thread Im interested in the easier questions of upper and lower bounds for those numbers.