Calculate acceptance ratio of Jacobian of split-merge RJMCMC

Question

I am keep studying the RJMCMC and want to ask question regarding the acceptance ratio of split/merge step of RJMCMC

The split/merge step, suggested by Richardson and Green (1997) is following for w_j, μ_j, θ_j for merging and splitting. Source : (https://academic.oup.com/jrsssb/article/59/4/731/7083042)

The acceptance ratio of above merge/splitting move, was proposed in the same paper (Richardson and Green, 1997)

My questions and confusions are below

1. They said (k+1) in the first line is from (k+1)!/k! of order statistics. However, the order of μ_j in this paper already defined as increasing order of value of μ_j. Then why we need that kind of factor of order statistics?

2. The first three lines of equation (11) is prior of each parameters and fourth line is proposal ratio. How to define/calculate proposal ratio of split/merge step of RJMCMC? In the split move from k-parameter, we have to choose one of k to split, so proposal probability should be 1/k. Also in merge move, we can choose k adjacent group to merge from (k+1) group of mixture. Then, r_m(x')/r_m(x)=(1/k)/(1/k)=1. Any contradiction in my thought process?

3. Can I find more detailed derived process of Jacobian calculation here of transformation from (w_j,μ_j,σ_j*^2,u1,u2,u3) to (w_j1,μ_j1,σ_j1^2,w_j2,μ_j2,σ_j2^2)?** It seems necessary to understand to fully understand that calculation to apply this approach to some different kind of model.

Thanks for reading my question. Apologies in advance if my question is unclear and doesn't make sens in some point.