Sorting permutations using a pop stack with a bypass
Abstract
We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously. In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.
Cite
@article{arxiv.2503.08285,
title = {Sorting permutations using a pop stack with a bypass},
author = {Lapo Cioni and Luca Ferrari and Rebecca Smith},
journal= {arXiv preprint arXiv:2503.08285},
year = {2025}
}
Comments
29 pages, 6 figures; this is the full version of the conference paper "Pop Stacks with a Bypass", in the Proceedings of the 13th edition of the conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings (GASCom 2024), Bordeaux, France, 24-28th June 2024, Electronic Proceedings in Theoretical Computer Science 403, pp. 73-78, arXiv:2406.16399v1