arXiv 2005.071872020-05-14math.CO

促销排序

Promotion Sorting

Colin Defant, Noah Kravitz

备注：: 16 pages, 4 figures

摘要：

Schützenberger的提升算子是一个广泛研究的双射，置换有限偏序集的线性扩张。我们引入了一个自然扩张$\partial$的操作，作用于所有的标签偏序集。我们证明了$\partial$的几个性质，特别是，我们表明，对于每一个标记$L$的$n$元素偏序集$P$，标记$\partial^{n-1}(L)$是一个线性扩展的$P$。因此，我们可以将$\partial$定义的动力系统视为将标签排序为线性扩展的排序过程。对于所有的$0\leq k\leq n-1$，我们的特点$n$元素偏序集$P$承认标签，至少需要$n-k-1$迭代$\partial$，以成为线性扩展。$k=0$涉及需要最大可能迭代次数才能排序的标记的情况;我们称这些标记为缠结的。我们明确列举了一个大类的偏序集，我们称之为膨胀根森林偏序集纠缠的标签。对于任意的有限偏序集，我们展示了如何枚举可排序的标签，这是标签$L$，使$\partial(L)$是一个线性扩展。

英文摘要：

Schützenberger's promotion operator is an extensively-studied bijection that permutes the linear extensions of a finite poset. We introduce a natural extension $\partial$ of this operator that acts on all labelings of a poset. We prove several properties of $\partial$; in particular, we show that for every labeling $L$ of an $n$-element poset $P$, the labeling $\partial^{n-1}(L)$ is a linear extension of $P$. Thus, we can view the dynamical system defined by $\partial$ as a sorting procedure that sorts labelings into linear extensions. For all $0\leq k\leq n-1$, we characterize the $n$-element posets $P$ that admit labelings that require at least $n-k-1$ iterations of $\partial$ in order to become linear extensions. The case in which $k=0$ concerns labelings that require the maximum possible number of iterations in order to be sorted; we call these labelings tangled. We explicitly enumerate tangled labelings for a large class of posets that we call inflated rooted forest posets. For an arbitrary finite poset, we show how to enumerate the sortable labelings, which are the labelings $L$ such that $\partial(L)$ is a linear extension.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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