Preimages under the bubblesort operator

TL;DR

This paper characterizes the preimages of permutations under the bubblesort operator, revealing explicit structures and properties of the associated permutation trees, and compares these findings to other sorting operators.

Contribution

It provides a detailed description of preimages under bubblesort, including explicit formulas and properties of the permutation trees, which was less understood compared to stacksort and queuesort.

Findings

Number of preimages is 2^{k-1} for k left-to-right maxima.

Properties of permutation trees related to maxima and suffixes.

Exact counts and average heights in the permutation trees.

Abstract

We study preimages of permutations under the bubblesort operator $\mathbf{B}$. We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators $\mathbf{S}$ (stacksort) and $\mathbf{Q}$ (queuesort). We describe explicitly the set of preimages under $\mathbf{B}$ of any permutation $\pi$ from the left-to-right maxima of $\pi$, showing that there are $2^{k-1}$ such preimages if $k$ is the number of these left-to-right maxima. We further consider, for each $n$, the tree $T_n$ recording all permutations of size $n$ in its nodes, in which an edge from child to parent corresponds to an application of $\mathbf{B}$ (the root being the identity permutation), and we present several properties of these trees. In particular, for each permutation $\pi$, we show how the subtree of $T_n$ rooted at $\pi$ is determined by the number of left-to-right maxima of $\pi$ and the length of the longest suffix of left-to-right maxima of $\pi$. Building on this result, we determine the number of nodes and leaves at every height in such trees, and we recover (resp. obtain) the average height of nodes (resp. leaves) in $T_n$.