Is there a $P$ time definable sequence of **succinct polynomial sized representation of balanced bipartite graphs** whose number of perfect matchings is a primorial?

For factorial a complete bipartite graph suffices.

Motivations:

The speed of growth of the function defining the sequence might capture prime gap information (for instance it might be able to provide lower bound on worst possible gap).

Since $n\#\approx 2^{O(n)}$ holds the sequence might be a sequence of planar graphs of sizes $O(n)$ in number of vertices and edges providing a way to count primes better.

The graphs perhaps satisfy symmetry as these objects usually exist for a reason (for complete balanced bipartites the symmetry group is canonical and perhaps for primorial as well there is a nice symmetry group and perhaps might hint at the construction of these family of bipartite graphs).

Input is $n$ but I seek succinct representation of a fixed family and the representation may be $polylog(n)$ sized expressing an exponentially larger balanced bipartite graph with perfect matching a primorial.

Succinct representation of balanced complete bipartite graphs could be a circuit of $2\lceil\log_2(n)\rceil$ input bits where the first half of the input bits represent one color and other half another color and the circuit trivially answers $1$ for all $n^2$ pairs of inputs to imply all pairs of vertices pairing the colors are connected. A boolean circuit for it is trivial as it always outputs $1$.

valueof $n$ rather than its encoded size (equivalently: unary input) is acceptable then there's a trivial answer which will disappoint you on all three points of the motivation. $\endgroup$