nRm is quite similar to nCm, except that it is circular combination. I don't have a single formulae for that. But I have a solution.
nCm = sum of (n/fi * |(n/fi)Q(m/fi)|) -- (1)
|nRm| = sum of(|(n/fi)Q(m/fi)|) -- (2)
where fi is a gcd(n,m)=g including 1 and g.
nQm is the subset of configurations from nRm, s.t. nQm does not have any symmetric ones.
The proof is by counting. So (2) is obvious. Only thing to remember is there for each element in Q there is one in R and viceversa.
Same type of counting works for (1) also.