Research in Enumerative Combinatorics

9972648调查者研究与连分式、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有关的列举组合学的研究课题。关于连分式的工作在两个方向上扩展了Flajolet的组合方法：第一，推广到一些著名的连分式，例如那些与Jacobi多项式的矩有关的连分式；第二，推广到很少被研究的连分式，称为“Lukasiewicz连分式”或“多连分式”。关于微分算子的工作用路径或有向图来解释它们在单项式上的作用，这些有向图可能代表一类非常一般的组合对象，其中包括那些由连分式产生的对象。在装箱算法方面的工作应用排列枚举的方法来研究某些装箱算法和双重装箱算法的行为。超几何级数理论提供了一种众所周知的方法来处理某些类型的恒等式，但有许多恒等式不是这种形式的，其中最有趣的是伯努利数和相关的序列。调查者研究这些身份，以期开发出处理它们的一般方法。调查者研究列举组合学的主题，这是一个数学领域，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。这一数学领域在计算机程序分析方面有很多应用，在统计学、物理学、化学和通信方面也有应用。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，有效地将各种大小的物体填充到容器中的方法的分析与某些类型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.