My research is in discrete probability, combinatorics and graph theory; in particular, extremal enumerative problems, set partitions, and the applications of combinatorial ideas to the study of phase transitions in statistical physics models and long-range correlations in discrete random structures. You can find a 2012 summary of my research here. Below are links to my papers (not necessarily the final versions); most of these can also be found at arXiv.org.

For links to journal versions of these papers, please go here; note that a Notre Dame netID is needed to access this page.

- Ordered partitions and the thinned exponential function (with J. Engbers and C. Smyth), not yet available.
- Total non-negativity of generalized Stirling matrices (with A. Pacurar)
- Independent set and matching permutations (with T. Ball, K. Hyry and K. Weingartner)
- Cutting lemma and Zarankiewicz's problem in distal structures (with A. Chernikov and S. Starchenko)
- Phase Coexistence for the Hard-Core Model on Z^2 (with A. Blanca, Y. Chen, D. Randall and P. Tetali),
*Combinatorics, Probability and Computing*,**28**(2019), 1-22. The data sets used in this paper are available here - The game of plates and olives (with T. Carroll),
*Electronic Journal of Combinatorics***26**(2019), P1.18 - Restricted Stirling and Lah number matrices and their inverses (with J. Engbers and C. Smyth),
*Journal of Combinatorial Theory Series A***161**(2019), 271-298 - Independent sets in the discrete hypercube (an expository article)
- The independent set sequence of some
families of trees (with J. Hilyard),
*Australasian Journal of Combinatorics***70**(2018), 236-252 - Extremal H-colorings of trees and 2-connected graphs (with J. Engbers),
*Journal of Combinatorial Theory Series B***122**(2017), 800-814 - On the independence ratio of distance graphs (with J. Carraher, S. Hartke, J. Radcliffe and D. Stolee),
*Discrete Mathematics***339**(2016), 3058-3072 - Asymptotic normality of some graph sequences,
*Graphs and Combinatorics***32**(2016), 639-647 - Phase coexistence and torpid mixing in the 3-coloring model on Z^d (with J. Kahn, D. Randall and G. Sorkin),
*SIAM Journal of Discrete Mathematics***29**(2015), 1223-1244 - Counting colorings of a regular graph,
*Graphs and Combinatorics***31**(2015), 629--638 - Combinatorially interpreting generalized Stirling numbers (with J. Engbers and J. Hilyard),
*European Journal of Combinatorics***43**(2015), 32--54 - Three tutorial lectures on entropy and counting (notes prepared to accompany a series of tutorial lectures given at the 1st Lake Michigan Workshop on Combinatorics and Graph Theory, Western Michigan University, March 15--16 2014), arXived
- Counting independent sets of a fixed size in graphs with given minimum degree (with J. Engbers),
*Journal of Graph Theory***76**(2014), 149--168 - Phase Coexistence and Slow Mixing for the Hard-Core Model on Z^2 (with A. Blanca, D. Randall and P. Tetali),
*Lecture Notes in Computer Science***8096**(*Proceedings of APPROX/RANDOM 2013*) (2013), 379--394. The data sets used in this paper are available here - Stirling numbers of forest and cycles (with Do Trong Thanh),
*Electronic Journal of Combinatorics***20**(2013), #P73 - Maximizing H-colorings of regular graphs,
*Journal of Graph Theory***73**(2013), 66--84 - H-coloring tori (with J. Engbers),
*Journal of Combinatorial Theory Series B***102**(2012), 1110-–1133 - The independent set sequence of regular bipartite graphs,
*Discrete Mathematics***312**(2012), 2881--2892 - H-colouring bipartite graphs (with John Engbers),
*Journal of Combinatorial Theory Series B***102**(2012), 726-–742 - Reverse Mathematics and infinite traceable graphs (with P. Cholak and R. Solomon),
*Mathematical Logic Quarterly***58**(2012), 18--28 - Two problems on independent sets in graphs,
*Discrete Mathematics***311**(2011), 2105--2112 - The multi-state hard core model on a regular tree (with Fabio Martinelli, Kavita Ramanan and Prasad Tetali),
*SIAM Journal of Discrete Mathematics***25**(2011), 894--916 - The number of independent sets in a graph with small maximum degree (with Yufei Zhao),
*Graphs and Combinatorics***27**(2011), 177--186. The java programs used in this paper (source files, compiled programs and documentation) are available here in a compressed (zipped) folder. - A threshold phenomenon for random independent sets in the discrete hypercube,
*Combinatorics, Probability and Computing***20**(2011), 27--51 - An upper bound for the number of independent sets in regular graphs,
*Discrete Mathematics***309**(2009), 6635--6640 - Matchings and Independent Sets of a Fixed Size in Regular Graphs (with Teena Carroll and Prasad Tetali),
*Journal of Combinatorial Theory Series A***116**(2009), 1219--1227) - Sampling independent sets in the discrete torus,
*Random Structures and Algorithms***33**(2008), 356--376 - Sampling 3-colourings of regular bipartite graphs,
*Electronic Journal of Probability***12**(2007), 481--497 - Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus (with Dana Randall),
*Proceedings of the Eighteenth Annual ACM--SIAM Symposium on Discrete Algorithms*(2007), 376--384 - Bounding the partition function of spin systems,
*Electronic Journal of Combinatorics***13**(2006), #R72 - Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs (with Prasad Tetali),
*Random Structures and Algorithms***28**(2006) 427--443 - Global connectivity from local geometric constraints for sensor network with various wireless footprints (with Raissa D'Souza, Cris Moore and Dana Randall),
*Proceedings of the Fifth International Conference on Information Processing in Sensor Networks*(2006), 19--26 - On weighted graph homomorphisms (with Prasad Tetali),
*DIMACS series in discrete mathematics and theoretical computer science***63**(*Graphs, Morphisms and Statistical Physics*) (2004), 97--104 - On phase transition in the hard-core model on Z^d (with Jeff Kahn),
*Combinatorics, Probability and Computing***13**(2004), 137--164 - Slow mixing of the Glauber Dynamics for the hard-core model on the Hamming cube (with Prasad Tetali),
*Proceedings of the Fifteenth Annual ACM--SIAM Symposium on Discrete Algorithms*(2004), 459--460 - Entropy and graph homomorphisms,
*Oberwolfach reports***1**(2004), 30--32 - On homomorphisms from the Hamming cube to Z,
*Israel Journal of Mathematics***138**(2003), 189--213 - Two problems involving the notion of phase transition (PhD thesis written under the direction of Jeff Kahn, Rutgers University, October 2002),
*ProQuest Dissertations and Theses*(2002)

**In preparation**

**Under revision**

**Submitted**

**2019**

**2018**

**2017**

**2016**

**2015**

**2014**

**2013**

**2012**

**2011**

**2002--2010**

- Ramanujan Graphs (Essay for Part III of Mathematical Tripos, University of Cambridge, June 1996)
- [4]^2 Tic-Tac-Toe is a draw
- Erdos's proof of Bertrand's postulate (an exposition of Erdos's elementary proof that there is always a prime between
*n*and*2n*) - Ultrafilters, with applications to analysis, social choice and combinatorics (an introduction to the basics of ultrafilters, with some applications)
- A topological approach to evasiveness (an introduction to the use of topological ideas in the study of evasive properties, based on a paper by Kahn, Saks and Sturtevant)
- Bregman's theorem and extensions (an exposition of Radhakrishnan's entropy proof of Bregman's theorem, and some results and conjectures around the theorem)

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