Algorithm to generate integer sets fulfills restrictions
You want to enumerate all integer partitions of $v$ into exactly $\ell$ parts, where each part is $\le b$.  Let $F(v,\ell,b)$ denote the set of all such partitions.

One can construct a recursive algorithm for this.  For instance, any such partition must include the number $b$ one, two, three, ..., or $\lfloor v/b \rfloor$ times, so:

$$F(v,\ell,b) = \bigcup_{i=0}^{\lfloor v/b \rfloor} F(v-ib,\ell-1,b-1) + b+\cdots + b,$$

where $F(\star)+b+\cdots+b$ represents the set of partitions (multisets) obtained by starting with each partition in $F(\star)$ and adding $i$ copies of the number $b$ to it.  Moreover, this is a disjoint union, so there is no need to use hashtables to eliminate duplicates.

However, the number of such partitions can be exponentially large in $v$, so in general the algorithm might be infeasibly expensive, unless $v$, $\ell$, or $b$ are sufficiently small.