The set of Schröder words (Schröder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schröder words as lattice paths. The resulting poset is called the Schröder pattern poset. We find closed formulas for the number of Schröder words covering/covered by a given Schröder word in terms of classical parameters of the associated Schröder path. We also enumerate several classes of Schröder avoiding words (with respect to the length), i.e. sets of Schröder words which do not contain a given Schröder word.Partially supported by INdAM-GNCS 2017 project: "Codici di stringhe e matrici non sovrapponibili". arXiv:1807.08648v1 [math.CO]
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 $T_n$.
We introduce a sorting machine consisting of k + 1 stacks in series: the first k stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes [10], which studies the case k = 1. Here we show that, for k = 2, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each k ≥ 3. Next we describe an optimal sorting algorithm, again for the case k = 2. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions. * G. C. and L. F. are members of the INdAM Research group GNCS; they are partially supported by IN-dAM -GNCS 2019 project "Studio di proprietá combinatoriche di linguaggi formali ispirate dalla biologia e da strutture bidimensionali" and by a grant of the "Fondazione della Cassa di Risparmio di Firenze" for the project "Rilevamento di pattern: applicazioni a memorizzazione basata sul DNA, evoluzione del genoma, scelta sociale".
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