Preimages under the bubblesort operator

Mathilde Bouvel¹ ; Lapo Cioni² ; Luca Ferrari²

¹Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
²Università degli Studi di Firenze

The electronic journal of combinatorics, Tome 29 (2022) no. 4

Résumé

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 $T_n$.

Détail

DOI : 10.37236/11390

Classification : 05A05, 05A15, 05C05
Mots-clés : sorting operators, average height of nodes, average height of leaves