How many distinct multisets (collections of legal chessmen) did you find? And was respecting bishop placement (but perhaps ignoring trapped bishops) observed?

No disrespect intended to James Propp. I too would persevere. Thus the suggested edit and the expressed hope for clarity. Like many other posters here, James "does not nail it the first time". That's all.

Indeed Mark, I hope clarity ensues. Given what I have seen from this poster, phrases like "with domain restricted appropriately" don't appear on the first pass.

I am suggesting an edit that includes the phrase "real numbers".

Of course I would now think of a 7 piece cage in a corner, giving perhaps an additional factor of 100 for 28 pieces

Many permutation matrices have that property. You might find similarity to an upper triangular matrix a more useful generalization wrt nilpotency.

Even wihout the consideration of fewer pieces, these 29 piece arrangements give at least twice as many positions as what you calculated for the 32 piece multiset. I think once the multisets are computed, we will find many fewer legal games than domotorp predicts.

I hope this recent edit affirms your faith. I suspect this forum can prove bounds to within a couple of orders of magnitude from the actual number.

Jose, thank you for your comments and the reference. If you would like, perhaps we could try a chat room. I think your computation of $\Omega(q^kn^2)$ above has an error; but I point out the inequality to emphasize that perfect numbers have lots of factors, and if I were more careful I would say Euler proved that odd perfect numbers are either powerful or mostly powerful, if they exist at all.

For the first million or so pairs of small primes, how do the stepsize and smallest positive member grow in an iteration? How long an iteration is needed before a cycle occurs? I imagine you will get some good conjectures out of such data. I foolishly predict that the proportion of increase will happen in C out of $\phi(p-q)$ cases for some small value of C.

I see I missed the title. However, it would be good to edit the question to include that trivial means the clone preserving $\rho$ has only projections.

It might help to clarify "trivial". The notion that makes the most sense to me is that the clone is trivial (only projection functions). It might also be that trivial means primal, but your examples don't suggest that. Can you relate primal to this notion of strongly rigid?