Preimages under the Queuesort algorithm
Lapo Cioni, Luca Ferrari

TL;DR
This paper explores the preimages of the Queuesort algorithm, providing recursive descriptions, enumeration results, and formulas involving ballot and Catalan numbers for specific permutation classes.
Contribution
It introduces a recursive method to find preimages of permutations under Queuesort and offers enumeration formulas for their counts, extending understanding beyond previous work on Stacksort.
Findings
All preimage cardinalities are possible except 3.
Exact counts for permutations with 0, 1, and 2 preimages.
Closed-form formula for permutations with a specific structure involving ballot and Catalan numbers.
Abstract
Following the footprints of what have been done with the algorithm Stacksort, we investigate the preimages of the map associated with a slightly less well known algorithm, called Queuesort. After having described an equivalent version of Queuesort, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0,1 and 2 preimages. Finally, we consider the special case of those permutations whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of : we determine a…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
