Research in Enumerative Combinatorics

9972648研究者研究与连分数，微分算子，与装箱算法相关的多面体以及伯努利数和相关序列的恒等式相关的枚举组合学的研究主题。 关于连分数的工作在两个方向上扩展了Flajolet的组合方法：第一，一些著名的连分数，例如与Jacobi多项式的矩相关的连分数，第二，连分数的一个很少研究的推广，称为“Lukasiewicz连分数”或“multicontinued分数”。 微分算子的工作解释他们的行动单项式的路径或有向图，这可能代表了一个非常普遍的一类组合对象，其中包括那些产生于连分数。 装箱算法的工作应用排列枚举的方法来研究某些装箱和对偶装箱算法的行为。 超几何级数理论给出了一种很好理解的方法来处理某些类型的恒等式，但也有许多恒等式不是这种形式，其中伯努利数和相关序列是最有趣的。 研究者研究这些恒等式，目的是找到处理这些恒等式的一般方法。研究者研究计数组合学的主题，计数组合学是一个数学领域，涉及计算各种运算可以执行的方式的数量，或者可以构造的特定类型的对象的数量。 这一领域的数学有许多应用程序的分析计算机程序，也有应用统计，物理，化学和通信。 这项工作扩展了解决几种不同类型的枚举问题的技术。 例如，其中一个主题表明，对用各种大小的对象有效地填充容器的方法的分析与某些种类的排列的枚举密切相关。

9972648The investigator studies study topics in enumerative combinatorics related to continued fractions, differential operators, polytopes associated with bin-packing algorithms, and identities for Bernoulli numbers and related sequences. The work on continued fractions extends Flajolet's combinatorial approach in two directions: first, to some well-known continued fractions, such as those associated with the moments of Jacobi polynomials, and second, to a little-studied generalization of continued fractions called "Lukasiewicz continued fractions" or "multicontinued fractions". The work on differential operators interprets their actions on monomials in terms of paths or directed graphs, which may represent a very general class of combinatorial objects, which include those arising from continued fractions. The work on bin-packing algorithms applies methods from permutation enumeration to study the behavior of certain bin-packing and dual bin-packing algorithms. The theory of hypergeometric series gives a well-understood way of dealing with certain types of identities, but there are many identies that are not of this form, of which the Bernoulli numbers and related sequences are among the most interesting. The investigator studies these identities, with a view to developing general ways to deal with them.The investigator studies topics in enumerative combinatorics, an area of mathematics that involves counting the number of ways that various kinds of operations can be performed, or the number of objects of a particular type that can be constructed. This area of mathematics has many applications to the analysis of computer programs, and also has applications to statistics, physics, chemistry, and communications. This work extends techniques for solving several different kinds of enumeration problems. One of the topics, for example, shows that the analysis of a method for efficiently filling containers with objects of various sizes is closely related to the enumeration of certain kinds of permutations.