Studies in Algebraic and Enumerative Combinatorics

该提案涉及与排列、多面体和偏序集(偏序集)相关的几个问题。有一些类型的凸多面体的组合性质，特别是它们的体积和Ehrhart多项式，将被研究。第一类多面体推广了一个广为人知的多面体，其体积(适当归一化)是1，2，…，n的交替排列的个数。第二类多面体由超单纯形的“半开”变体及其一些推广组成。根据排列的下降次数和超出次数，排列的计数有一个令人惊讶的联系。最后一类是偏序集上鲜为人知的赋值多面体。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划分格上的作用。由此得到的偏序集是分割格的一种商，是一个超可解的EL-shell型格，它有望具有许多有趣的附加性质。最近关于区间序的工作建议将这些结果推广到标记区间序，这是PI以前引入的一个概念。他希望找到最近研究的概念的“有标记的类似物”，如上升序列、避免某种条形图案的排列和斯托伊门诺对合(或规则的线性化弦图)。一个相关的问题是找到一种理论，统一某些类别的标记对象和未标记对象之间的联系。2007年，K.Saito证明了一个关于树的有趣结果，它推广了Niven和De Bruijn(独立地)关于排列的一个定理。齐藤猜想，他的结果可以推广到所有的二部图。PI将首先使用欧拉偏序集CD指数理论中的技巧来证明齐藤猜想。PI将继续与R.Du一起研究元素在两个循环的乘积的循环中的分布，最初的灵感来自M.Bona的一个猜想。特别是，他将考虑几个公开的问题，这些问题是他以前与DU合作时产生的。多面体、排列和偏序集在整个数学中无处不在。关于它们有许多深刻而优雅的定理，但这些对象的例子是有限的，对于它们的基本不变量有明确的描述。新的例子将为应用程序打开大门，为描述不变量所涉及的数学提供新的启示，并提供以前不相关的对象之间的新联系。区间序在社会学、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具有类似的应用。斋藤的猜想暗示了一个经过充分研究的几何概念的泛化，这可能具有广泛的适用性。

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.