In this paper, we study merging-free partitions and run-sorted permutations. We give a combinatorial proof of a conjecture of Nabawanda, Rakotondrajao, and Bamunoba. 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 that the distribution of right-to-left minima is the shifted distribution of the Stirling numbers of the second kind. We also prove that the number of non-crossing merging-free partitions is a power 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 the number of runs.