Enumerative combinatorics and applications

我的研究项目主要涉及数学和科学的其他分支产生的各种计数问题，以及与标记和非标记枚举相关的适当数学和组合工具的开发。我的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(多项式计数、迈耶展开)、组合化学、拓扑图论等。例如，我正在进行的研究问题包括：-计算由迈耶对气体模型的配分函数展开而产生的图形不变量。--2-边连通图的勒让德变换的组合解释。-各类凸多项式的计数和渐近展开。-根据其对称性对苯类分子进行分类。-不含k33-细分的射影平面图和环形图的特征。第二--各种树状结构、仙人掌和2-树的计数和分类。我还参与了更多的理论工作，在那里，列举组合学的基本工具和方法得到了发展和加强，特别是物种组合理论的工具和方法。这种方法的目标之一是根据结构的稳定基团对结构进行分类，特别是对于由函数方程定义的物种。计算机代数的使用在这项工作中发挥了核心作用。我在这一领域的一些研究活动包括：*将两极网络替换为图形结构边缘的方法的扩展，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.