Enumerative combinatorics and applications

我的研究项目涉及各种各样的枚举问题，这些问题主要来自数学和科学的其他分支，也涉及与标记和非标记枚举相关的适当的数学和组合工具的发展。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（多项式枚举、Mayer展开）、组合化学、拓扑图理论等。我正在进行的研究问题包括，例如：-计算由Mayer对气体模型的配分函数展开引起的图不变量。-基于2边连通图的Legendre变换的组合解释。各种凸多项式的枚举和渐近展开式。-根据苯类分子的对称性进行分类。-无k33细分的投影平面图和环面图的表征。-列举和分类各种树状结构，仙人掌和2-树。我还参与了一项更理论化的工作，其中枚举组合学的工具和方法得到了发展和加强，特别是那些物种组合理论。这种方法的目标之一是根据它们的稳定群对结构进行分类，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一努力中起着核心作用。我在这一领域的一些研究活动包括：将2极网络替换为图形结构边缘的方法的扩展，以及特定种类（例如定向环和多边形）的加法公式的计算。

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.