Combinatorics is the study of finite structures. In particular, combinatorics is often interested in the existence, construction, enumeration, and/or optimization of certain types of finite structures.
Miklos Bona is an enumerative combinatorialist. He likes to count all sorts of objects, but has studied permutations, trees, and partially ordered sets the most often, using both analytic and algebraic methods. His additional interests are in sorting algorithms and mathematical biology.
Vince Vatter is interested generally in the combinatorics of relational structures, particularly permutations and graphs.
The research interests of Andrew Vince lie in combinatorics, graph theory, discrete geometry, and fractal geometry. An example of a self-similar polygonal tiling from a recent paper appears below. Other interests include hiking and bicycling.
The leading image shows an example of a self-similar polygonal tiling.