The method of switchings is a standard tool for enumerative and probabilistic applications in combinatorics. In its simplest form, it analyses a relation between two sets to estimate the ratio of their sizes. In a more complicated setting, there is a family of sets connected by some relations. By bounding properties of the relations, bounds can be inferred on the relative sizes of the sets. In ...

My research interests lie in the areas of enumerative and algebraic combinatorics. In particular, I am studying pattern avoidance for affine permutations and its applications. For many years, many mathematicians have studied pattern avoidance in permutations. Sometimes the questions are as simple as, ”How many permutations avoid a given set of patterns?” For example, the number of permutations ...

In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X ; (b) the observation of...

My research has touched many areas of combinatorics, including enumerative, algebraic, spectral, extremal and probabilistic. The projects below include finding the stationary distribution of a random coloring procedure, pursuing new directions in the math of juggling, working with undergraduates on Ramsey theory, and studying a pair of topics from spectral graph theory. I have mentored undergra...

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this...

We give a lattice path interpretation for totally symmetric self-complementary plane partitions. This is a first step in solving the long standing problem of enumerating such plane partitions. Another outstanding problem in enumerative combinatorics is the search for a bijection between alternating sign matrices and totally symmetric self-complementary plane partitions. From the lattice path in...

In this paper we want to study combinatorics of the type B permutations and in particular the join statistics crossings, excedances and the number of negative entries. We generalize most of the results known for type A (i.e. zero negative entries) and use a mix of enumerative, algebraic and bijective techniques. This work has been motivated by permutation tableaux of type B introduced by Lam an...

We present a theorem-proving experiment performed with a computer algebra system. It proves a conjecture about the general pattern of the generating functions counting rooted maps of given genus. These functions are characterized by a complex non-linear differential system between generating functions of multi-rooted maps. Establishing a pattern for these functions requires a sophisticated indu...

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2ρ−n, where g ≈ 0.00909941 and ρ−1 ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n...

Monotone path polytopes arise as a special case of the construction of ber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement ca...