My research is in enumerative and algebraic combinatorics. I like to count things such as permutations, partitions, and graphs, often with the help of symmetric functions and other algebraic objects. My favorite questions in this area are asymptotic: for instance, I have proved asymptotic results about the number of cyclic permutations with a given descent set, and I have proved a conjecture on the asymptotic number of split graphs that are "balanced". I also do work in permutation patterns, especially on growth rates of permutation classes.
My main research focus right now is my joint project with Neal Madras, my postdoctoral supervisor. We are studying pattern avoidance in "periodic permutations": bijections from ℤ to ℤ that repeat with a finite period and meet certain other conditions. We have some interesting results here, and we conjecture that the growth rate of a pattern-avoiding class of periodic permutations is equal to that of the corresponding ordinary permutation class.
Here are my research papers:
- Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019): #P2.42.
- On the centrosymmetric permutations in a class, Australas. J. Combin., 74 (2019): 423–442.
- Exact and asymptotic enumeration of cyclic permutations according to descent set, with S. Elizalde, J. Combin. Theory Ser. A, 165 (2019): 360–391 (free version on arXiv).
- Combinatorial species and graph enumeration (undergraduate thesis), with A. Hardt, P. McNeely, and T. Phan, 2013. A concise expository introduction to the theory of combinatorial species. Includes original results on the enumeration of certain kinds of graphs; also includes Sage code for combinatorial species.