Complexity of Combinatorial Sequences

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Enumerative combinatorics is the branch of mathematics that deals with counting the number of objects, such as sets, permutations, matrices,..., satisfying certain prescribed conditions. In the past several decades enumerative combinatorics has been one of the most rapidly developing areas of mathematical research, with numerous connections and applications. The area pushed away traditional boundaries and joined forces with a variety of fields in mathematics and beyond, ranging from computer science to statistical physics. The investigator will undertake a profound study of integer sequences using a broad and powerful range of mathematical tools. The central sequences that will be studied arise in enumerative combinatorics, involving counting pattern-avoiding permutations, graphs ina hereditary property, and closed walks in Cayley graphs. The investigator will involve students at all levels in his research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.