Enumerative combinatorics and applications

点击翻译按钮获取中文摘要

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.