Preimages under the Queuesort algorithm

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 $\pi$ whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of $\pi$: we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers.