关于巢多面体面上的广义翻转序
Generalised flip order on the faces of nestohedra
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中文总结 AI 辅助
研究某些巢多面体族面上洗牌积与广义翻转序的关系，通过对编码对象的操作定义广义翻转序，将其从顶点扩展到所有面，还与其他序比较并给出基于（广义）逆序的特征描述。
中文摘要 AI 辅助
排列和二元平面有根树上的经典洗牌积（即在置换多面体和关联多面体的顶点上）分别可以用弱布劳威尔序和塔马里序中的区间来描述。帕拉西奥斯和龙科将这些积及其区间描述扩展到了满射和平面有根树（即在置换多面体和关联多面体的所有面上）。本文给出了这一现象的广泛推广。我们表明，某些巢多面体族面上的洗牌积对于广义翻转序允许一个区间描述，广义翻转序是通过对编码它们的树状组合对象进行基本的分裂和融合操作在巢多面体的面上定义的偏序。广义翻转序将巴纳德和麦康维尔的翻转序从顶点扩展到了巢多面体的所有面。我们进一步将其与德尔门吉安 - 霍尔韦格 - 皮劳德的面弱序和龙科的广义塔马里序进行比较，并根据（广义）逆序给出其特征描述。
英文摘要
Classical shuffle products on permutations and binary planar rooted trees (i.e., on the vertices of permutohedra and associahedra) admit descriptions in terms of intervals in the weak Bruhat order and the Tamari order, respectively. Palacios and Ronco extended these products as well as their interval description to surjections and planar rooted trees (i.e. on all faces of permutohedra and associahedra). In this article, we present a broad generalisation of this phenomenon. We show that the shuffle product on faces of certain families of nestohedra admits an interval description with respect to the generalised flip order, a partial order defined on the faces of nestohedra through elementary splitting and fusion operations on the tree-like combinatorial objects encoding them. The generalised flip order extends the flip order of Barnard and McConville from vertices to all faces of nesthedra. We further compare it with the facial weak order of Dermenjian-Hohlweg-Pilaud and the generalised Tamari order of Ronco, and we provide its characterisation in terms of (generalised) inversions.
