Studies in Algebraic and Enumerative Combinatorics

代数和枚举组合学研究

• 批准号: 1068625
• 负责人: Richard Stanley
• 金额: $ 59.86万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068625&HistoricalAwards=false
• 关键词: Studies; Algebraic; Enumerative; Combinatorics

The proposal concerns several problems related to permutations, polytopes, and partially ordered sets (posets). There are a number of classes of convex polytopes whose combinatorial properties, especially their volumes and Ehrhart polynomials, will be investigated. The first of these polytopes generalizes a well-understood polytope whose volume (suitable normalized) is the number of alternating permutations of 1,2,...,n. The second class of polytopes consists of "half-open" variants of a hypersimplex and some generalizations thereof. There is a surprising connection with the enumeration of permutations according to their number of descents and number of excedances. The final class consists of the poorly understood valuation polytopes of posets. The PI will consider some posets related to group actions on other posets, the primary example being the action of the symmetric group on the lattice of partitions of a set. The resulting poset, a kind of quotient of the partition lattice, is a supersolvable and EL-shellable lattice, and it promises to have a host of interesting additional properties. Recent work on interval orders suggests extending these results to marked interval orders, a concept previously introduced by the PI. He hopes to find "marked analogues" of such recently studied concepts as ascent sequences, permutations avoiding a certain barred pattern, and Stoimenow involutions (or regular linearized chord diagrams). A related problem is to find a theory unifying the connection between certain classes of labelled and unlabelled objects. In 2007 K. Saito proved an intriguing result about trees that generalizes a theorem of Niven and de Bruijn (independently) about permutations. Saito conjectured that his result could be extended to all bipartite graphs. The PI will try to prove Saito's conjecture, first using techniques from the theory of the cd-index of an Eulerian poset. The PI will continue research with R. Du on the distribution of elements in the cycles of a product of two cycles, inspired originally by a conjecture of M. Bona. In particular, he will consider several open problems arising from his previous work with Du.Polytopes, permutations, and posets are pervasive throughout mathematics. There are many deep and elegant theorems concerning them, but examples of these objects are limited for which there exist explicit descriptions of their fundamental invariants. New examples would open doors to applications, shed new light on the mathematics involved in the description of the invariants, and provide new connections between previously unrelated objects. Interval orders have numerous applications to such areas as sociology and psychology. An expansion of the theory of marked interval orders may have similar applications. Saito's conjecture hints at a generalization, that could have broad appicability, of a well-studied geometric concept.

该建议涉及到几个问题有关的排列，多面体，偏序集（偏序集）。 有一些类的凸多面体的组合性质，特别是它们的体积和Ehrhart多项式，将被调查。这些多面体中的第一个推广了一个很好理解的多面体，其体积（适当的归一化）是1，2，.的交替排列的数量，n.第二类多面体由超单形的“半开”变体及其一些推广组成。根据下降数和超出数来列举排列，这两者之间有一种令人惊讶的联系。 最后一类是偏序集的赋值多面体。 PI将考虑一些与其他偏序集上的群作用相关的偏序集，主要的例子是对称群在集合的分区格上的作用。由此产生的偏序集，一种商的划分格，是一个超可解和EL-壳格，它承诺有一系列有趣的额外的性质。 最近的工作间隔订单建议延长这些结果，以显着的间隔订单，以前介绍的PI的概念。他希望找到最近研究的概念的“标记类似物”，如上升序列，避免某种禁止模式的排列，以及Stoimenow对合（或规则线性弦图）。一个相关的问题是要找到一个理论统一之间的联系某些类别的标签和非标签对象。2007年K. Saito证明了一个有趣的结果，推广了Niven和de Bruijn（独立）关于置换的定理。Saito证明了他的结果可以推广到所有的二部图。PI将尝试证明齐藤的猜想，首先使用欧拉偏序集的cd指数理论的技术。 PI将继续研究R。Du关于两个圈的乘积的圈中元素的分布，最初受到M.博纳.特别是，他将考虑几个开放的问题所产生的，他以前的工作与杜。多面体，排列，偏序集是普遍存在的整个数学。有许多深刻而优雅的定理关于他们，但这些对象的例子是有限的，其中存在明确的描述，其基本不变量。新的例子将为应用打开大门，为描述不变量所涉及的数学提供新的线索，并在以前不相关的对象之间提供新的联系。区间序列在社会学和心理学等领域有着广泛的应用。标记区间序理论的扩展可能有类似的应用。斋藤的猜想暗示了一个普遍化，这可能具有广泛的适用性，一个良好的研究几何概念。