Counting Pop-Stacked Permutations in Polynomial Time

Permutations that can be sorted greedily by one or more stacks having various constraints have been studied by a number of authors. A pop-stack is a greedy stack that must empty all entries whenever popped. Permutations in the image of the pop-stack operator are said to be pop-stacked. Asinowki, Banderier, Billey, Hackl, and Linusson recently investigated these permutations and calculated their number up to length 16. We give a polynomial-time algorithm to count pop-stacked permutations up to a fixed length and we use it to compute the first 1000 terms of the corresponding counting sequence. With the 1000 terms, we apply a pair of computational methods to prove some negative results concerning the nature of the generating function for pop-stacked permutations and to empirically predict the asymptotic behavior of the counting sequence using differential approximation.