  In this paper, we study merging-free partitions with their canonical forms
and run-sorted permutations. We give a combinatorial proof of the conjecture
made by Nabawanda et al. We describe the distribution of the statistics of runs
and right-to-left minima over the set of run-sorted permutations and we give
the exponential generating function for their joint distribution. We show the
number of right-to-left minima is given by the shifted distribution of the
Stirling number of the second kind. We also prove that the non-crossing
merging-free partitions are enumerated by powers of 2. We use one of the
constructive proofs given in the paper to implement an algorithm for the
exhaustive generation of run-sorted permutations by number of runs.
