Some distributions on increasing and flattened permutations

We examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide an exponential generating function for its corresponding distribution and popularity. Additionally, we present one-to-one correspondences between these permutations and some classes of simpler combinatorial objects.