Problems in Enumerative Combinatorics and Applications

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This research project is on enumerative and algebraic combinatorics, a fast developing area in contemporary mathematics. The PI explores a series of interrelated enumeration problems on various combinatorial structures, including permutations, matchings, set partitions, integer sequences, general graphs, and fillings of Ferrers boards and other polyominoes. Throughout mathematics a common and successful approach to studying some objects of interest is to study functions on those objects. In combinatorics such functions are often called combinatorial statistics. This research project emphasizes on the algebraic properties of combinatorial statistics and their connections to other branches of mathematics. The major topics include(1) developing an algebraic theory for the interpolating families between equidistributed combinatorial statistics,(2) studying descent sets and descent polynomials for various combinatorial structures and their relations to the emerging field of one-dependent determinantal point processes,(3) characterizing the crossing and nestings in combinatorial structures with applications in graph optimizations and computational mathematical biology.The proposed research is in the center of combinatorics, which is a branch of mathematics concerning the existence, enumeration, analysis, and optimization of discrete structures. The investigator uses a combined algebraic and combinatorial approach to investigate properties and statistics of many discrete structures, and brings coherence and unity to the discipline of combinatorics. Results of the proposed research have connections and applications to such classical areas of mathematics as algebra, group representation theory, number theory, and probability theory, as well as more applied subjects, notably computer science, graph encoding and drawing, VLSI design, and mathematical biology.