Lower bound on the running time of Pop-Stack Sorting on a random permutation

Pop-Stack Sorting is an algorithm that takes a permutation as an input and sorts its elements. It consists of several steps. At one step, the algorithm reads the permutation it has to process from left to right and reverses each of its maximal decreasing subsequences of consecutive elements. It terminates at the first step that outputs the identity permutation. In this note, we answer a question of Defant on the running time of Pop-Stack Sorting on the uniform random permutation $\sigma_n$. More precisely, we show that there is a constant $c > 0.5$ such that asymptotically almost surely, the algorithm needs at least $cn$ steps to terminate on $\sigma_n$.