Flip sort and extremal cases of pop-stack sorting

Pop-stack sorting is a natural sorting procedure and a fascinating process to analyse. It finds its roots in the seminal work of Knuth on sorting algorithms and permutation patterns. We present several results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we represent the "2-pop-stack sortable permutations" by lattice paths to prove conjectures raised by Pudwell and Smith, and we characterize some families of permutations related to the image of the pop-stack sorting and to its "worst case".