In my research, 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 results about how many “very large” cyclic permutations have a given descent set, and I have proved a conjecture on the number of “very large” split graphs—in both cases, “very large” means we are taking the limit as the size of the structure is going to infinity. In much of this work I have applied analysis and probability to answer questions about combinatorics. Much of my work is in permutation patterns, especially on growth rates of permutation classes.

Here is my research statement.

Here is my list of publications, works in progress, and unpublished papers:

- J. J. Fang, Z. Hamaker, and J. M. Troyka, “On pattern avoidance in matchings and involutions”, arXiv:2009.00079 (submitted for publication).
- J. M. Troyka and Y. Zhuang, “Fibonacci numbers, consecutive patterns, and inverse peaks”, arXiv:2109.14774 (submitted for publication).
- J. M. Troyka and Y. Zhuang, “Permutation statistics on clusters”, in progress.
- J. M. Troyka, “Growth rates of permutations with a given descent set”, in progress.
- N. Madras and J. M. Troyka, “Bounded affine permutations II. Avoidance of decreasing patterns”,
*Ann. Comb.*(2021): https://doi.org/10.1007/s00026-021-00553-4. - N. Madras and J. M. Troyka, “Bounded affine permutations I. Pattern avoidance and enumeration”,
*Discrete Math. Theor. Comput. Sci.***22**(2) (2021): #1. - J. M. Troyka, “Period mimicry: A note on the (–1)-evaluation of the peak polynomials”, arXiv:1907.06681 (2019).
- J. M. Troyka, “Split graphs: Combinatorial species and asymptotics”,
*Electron. J. Combin.***26**(2019): #P2.42. - J. M. Troyka, “On the centrosymmetric permutations in a class”,
*Australas. J. Combin.***74**(2019): 423–442. - S. Elizalde and J. M. Troyka, “Exact and asymptotic enumeration of cyclic permutations according to descent set”,
*J. Combin. Theory Ser. A***165**(2019): 360–391 (free version on arXiv). - A. Hardt, P. McNeely, T. Phan, and J. M. Troyka, “Combinatorial species and graph enumeration” (undergraduate senior thesis), arXiv:1312.0542 (2013).
*A concise expository introduction to the theory of combinatorial species.* - A. Hardt and J. M. Troyka, “Restricted symmetric signed permutations”,
*Pure Math. Appl.***23**(2012): 179–217.