For a 3SAT formula with $ n$ variables and $ m$ clauses, I am interested in counting the number of isomorphic formulas (isomorphic in the sense that they are logically equivalent *and* have the same number of variables and clauses). I assume that we can invert the sense of any variable (swap all $ x_i$ with $ \lnot x_i$ ) and permute any of the variables. The first should give $ 2^n$ and the second should give $ n!$ .

Is this all of the isomorphisms, or am I overlooking anything?

If this is correct, then I assume that the number of distinct 3SAT formulas with $ n$ variables and “distinct clauses” is equal to $ \frac{2^m}{2^nn!}$ , where $ m$ is the number of 3-clauses over $ n$ variables in which all variables are distinct. I assume that $ m=2n(2n-2)(2n-4)$ . Does that sound right?