Preimages under the bubblesort operator
Mathilde Bouvel, Lapo Cioni, Luca Ferrari

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 . We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators (stacksort) and (queuesort). We describe explicitly the set of preimages under of any permutation from the left-to-right maxima of , showing that there are such preimages if is the number of these left-to-right maxima. We further consider, for each , the tree recording all permutations of size in its nodes, in which an edge from child to parent corresponds to an application of (the root being the identity permutation), and we present several properties of these trees. In particular, for each permutation , we show how the subtree of rooted at is determined by the number of…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · semigroups and automata theory · Algorithms and Data Compression
