Merging-Free Partitions and Run-Sorted Permutations

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.