Research in Enumerative Combinatorics

调查研究的问题，在枚举组合相关的排列，对称函数，和格路径。研究者的方法之一是将理查德·斯坦利的P-划分理论及其推广应用于对称群和相关群的下降代数的研究。还研究了与下降排列计数有关的其他几个问题。调查员还研究了两个堆栈排序排列的推广导致一个精确的函数细化公式的西方（证明Zeilberger）计数这些排列，和几个问题，涉及枚举的lattice routes.Enumerative组合涉及计数的数学对象与给定的属性。它可以应用于计算机科学、物理学和化学，以及其他数学领域，如代数、概率和统计、拓扑学、分析和数论。调查员的长期目标是发展，扩展和系统化的方法枚举组合，和研究的问题将有助于实现这一目标。

The investigator studies problems in enumerative combinatorics related to permutations, symmetric functions, and lattice paths. One of the investigator's approaches is to apply Richard Stanley's theory of P-partitions and its generalizations to the study of descent algebras of symmetric and related groups. Several other problems related to the enumeration of permutations by descents are also studied. The investigator also studies a generalization of two-stack-sortable permutations leading to a symmetric-function refinement of the formula of West (proved by Zeilberger) for counting these permutations, and several problems involving the enumeration of lattice paths.Enumerative combinatorics involves counting mathematical objects with given properties. It has applications to computer science, physics, and chemistry, as well as to other areas of mathematics, such as algebra, probability and statistics, topology, analysis, and number theory. The investigator's long-range goal is to develop, extend, and systematize the methods of enumerative combinatorics, and the problems studied will contribute towards this goal.