论文标题
$ s $ - 企业订单和$ S $ -Permutahedra I：组合和晶格结构
The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure
论文作者
论文摘要
这是介绍$ s $ weak订单和$ s $ -permutahedra概念的一系列论文的第一贡献，这些概念是某些由一系列非负整数$ s $索引的离散对象。在第一篇论文中，我们纯粹专注于$ s $ weak Order的组合和晶格结构，这是对某些减少树木的部分订单，这些树木概括了对排列的经典弱点。特别是，我们表明$ S $ weak Order是一个半分配和一致性统一的晶格，它概括了已知的结果，以置于排列的经典弱顺序。 将$ s $ weak的订单限制为某些树木会产生$ s $ tamari lattice，$ s $ tamari lattice，这是一种概括古典塔玛里晶格的sublattice。我们表明，当$ s $ s $ weak Order的商晶格可以获得$ s $ -tamari晶格，当$ s $没有零零时，可以获得$ s $ weak订单，并表明$ s $ tamari lattices（用于任意$ s $）是$ $ n是$ n是$ n是$ n是$ n是$ n to $ n是$ n to $ n是$ν$ n $ν$ tamari-tamari-tamari lattices ofprévilille-tamari lattices ofpréville-ville-cliville-claterle-lattices。 $ s $ weak订单的基本几何结构将在本文的续集中研究，我们介绍了$ s $ permutahedra的概念。
This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate purely on the combinatorics and lattice structure of the $s$-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the $s$-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the $s$-weak order to certain trees gives rise to the $s$-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the $s$-Tamari lattice can be obtained as a quotient lattice of the $s$-weak order when $s$ has no zeros, and show that the $s$-Tamari lattices (for arbitrary $s$) are isomorphic to the $ν$-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the $s$-weak order will be studied in a sequel of this paper, where we introduce the notion of $s$-permutahedra.
