Enumerative combinatorics and applications

我的研究计划主要涉及数学和科学的其他分支所产生的各种枚举问题，以及与标记和未标记枚举相关的适当数学和组合工具的开发。 应用领域包括数学分析（正交多项式和其他特殊函数的组合模型）、计算机科学（数据结构）、统计力学（多角枚举、Mayer展开）、组合化学、拓扑图论等。 我正在进行的研究问题包括，例如：-计算图形不变量所产生的迈耶的扩展配分函数的气体模型。- 勒让德变换在2-边连通图中的组合解释。- 各类凸多项式的计数与渐近展开。- 苯型分子根据对称性的分类。- 不含k33-剖分的射影平面图和环面图的刻画。 - 各类树状结构、仙人掌和2-树的计数和分类。 我还参加了更多的理论努力，在枚举组合学的工具和方法的开发和增强，特别是那些物种的组合理论。 这种方法的目标之一是根据其稳定剂组，特别是由功能方程定义的物种的结构的分类。计算机代数的使用在这一努力中起着核心作用。 我在这方面的一些研究活动包括：将2-pole网络替换为图形结构的边缘的方法的扩展，以及特定物种的加法公式的计算，例如定向循环和多边形。

My research program deals with various enumerative problems arising mainly from other branches of mathematics and science and also with the development of appropriate mathematical and combinatorial tools related to labelled and unlabelled enumeration.  The areas of application include mathematical analysis (combinatorial models for orthogonal polynomials and other special functions), computer science (data structures), statistical mechanics (polyomino enumeration, Mayer expansions), combinatorial chemistry, topological graph theory, etc.    My ongoing research problems include, for example: - Computation of graph invariants arising from Mayer's expansion of partition functions for gas models. - A combinatorial interpretation of the Legendre transform in terms of 2-edge-connected graphs. - Enumeration and asymptotic expansion of various classes of convex polyominoes. - Classification of benzenoid molecules according to their symmetries. - Characterization of projective-planar and toroidal graphs with no k33-subdivisions.    - Enumeration and classification of various classes of tree-like structures, cacti and 2-trees.   I also participate in a more theoretical effort, where the  tools and methods of enumerative combinatorics are developed and enhanced, particularly those of the combinatorial Theory of Species.   One of the objectives of this approach is the classification of structures according to their stabilizer groups, in particular for species which are defined by functional equations. The use of computer algebra plays a central role in this endeavour.   Some of my research activities in this area include:  The extension of the methods for the substitution of 2-pole networks into the edges of graphical structures, and the computation of addition formulas for  particular species for example oriented cycles and polygons.