An automaton consists of a finite set of states and a collection of functions from the set of states to itself. We call an automaton synchronizing if there is a word (that is, a sequence of functions) that maps all states onto the same state. Cerný's conjecture on the length of the shortest such word is probably the most famous open problem in automata theory. We consider the closely related question of determining the minimum length of a word that maps k states onto a single state.

For synchronizing automata, we have improved the upper bound on the minimum length of a word that sends some triple to a single state from 0.5n^2 to 0.19n^2. I will discuss this result and some related results, including a generalisation of this approach this to an improved bound on the length of a synchronizing word for 4 states and 5 states.

This is joint work with Robert Johnson.

Graphs with extremal properties are often found within the class of threshold graphs. Threshold graphs can be obtained from any graph by repeated application of the Kelmans transformation. This transformation has been shown to have a monotonic effect on the number of spanning substructures of a graph, including the number of spanning trees. In this talk, we will show that the Kelmans transformation decreases the number of spanning pseudoforests of a graph. This result implies that there exists a threshold graph on n vertices and e edges with the minimum number of spanning pseudoforests.

Determining whether an arbitrary lattice is the face lattice of a convex polytope is an old and difficult problem known as the algorithmic Steinitz problem. Such a question is just one of many that can be addressed by studying the realization space of a polytope. In this talk I will introduce the concept of a realization space and describe several models used to study these spaces. I will highlight a new model, which represents a polytope by its slack matrix, and provides an algebraic approach to answering realization space questions. We will see how this new model is connected to previously studied models (including a Grassmannian model and one using Gale diagrams), as well as how these connections can be exploited to answer the algorithmic Steinitz problem for arbitrary lattices that were intractable by previous methods.

This is joint work with João Gouveia and Antonio Macchia.

Macdonald polynomials have long been hailed as a breakthrough in algebraic combinatorics as they simultaneously generalize both Hall-Littlewood and Jack symmetric polynomials. The nonsymmetric Macdonald polynomials E_a(X;q,t) are a further generalization which contain the symmetric versions as special cases. When specialized at t =0 the nonsymmetric Macdonald polynomials were shown by Bogdon and Sanderson to arise as characters of affine Demazure modules, which are certain truncations of highest weight modules. In this talk, I will describe a type A combinatorial crystal which realizes the affine Demazure module structure and recovers the results of Bogdon and Sanderson crystal-theoretically and also defines a filtration of these modules by finite Demazure modules. Thus, we show the specialized Macdonald polynomials are graded sums of finite Demazure characters or Key polynomials. As a consequence, we derive a new combinatorial formula for the Kostka-Foulkes polynomials. 

This is joint work with Sami Assaf.

How many d-regular k-uniform hypergraphs are there on n vertices? We provide an asymptotic formula covering a wide range of parameters, including regular k-uniform hypergraphs of density up to c/k for 2 < k < n^(1/10). We shed light on the perturbation method, originally applied to the graph enumeration problem by Liebenau and Wormald.

This is joint work with Anita Liebenau and Nick Wormald.

For n ≥ s > r  ≥ 1 and k ≥ 2, write n → (s)_k^r if every colouring with k colours of the complete r-uniform hypergraph on n vertices has a homogeneous subset of size s. Improving upon previous results by Axenovich et al. (2014) and Erdős et al. (1984) we show that if r ≥ 4 and  n → (s)_k^r  then 2^n \(s+1)_{k+4}^{r+1}. This yields a small increase for some of the known lower bounds on multicolour hypergraph Ramsey numbers.

This is a joint work with Bruno Jartoux, Chaya Keller and Shakhar Smorodinsky.

Symmetry in a graph G can be measured by investigating possible automorphisms of G. One way to do this is to color the vertices of G in such a way that only the trivial automorphism can preserve the color classes. If such a coloring exists with d colors, G is said to be d-distinguishable. The smallest d for which G is d-distinguishable is its distinguishing number. Another measure of symmetry is to consider subsets S ⊆ V(G) such that the only automorphism that fixes the elements of S pointwise is the trivial automorphism. Such sets S are called determining sets for G and the determining number of G is the size of a smallest determining set. Though the two parameters were introduced separately, relationships exist. In particular, a determining set is a useful tool for finding the distinguishing number. In this talk we'll investigate both parameters in the setting of simple graphs achieved by applying the traditional Mycielskian and generalized Mycielskian constructions. The traditional Mycielskian construction was introduced by Mycielski in 1955 to prove that there exist triangle-free graphs with arbitrarily large chromatic number.

This is joint work with Debra Boutin, Sally Cockburn, Lauren Keough, Sarah Loeb, and Puck Rombach.

Both the noncrossing partition/Kreweras lattice and parking functions are well-studied objects in combinatorics. In 1997 Richard Stanley found a bijection between the maximal chains in the lattice and parking functions. We discuss what happens under the bijection when we restrict to certain induced sublattices.

This is joint work with the following colleagues:  Shreya Ahirwar, Mount Holyoke College;  Susanna Fishel, Arizona State University; Parikshita Gya, Mount Holyoke College; Pamela E. Harris, Williams College; Nguyen (Emily) Pham, Mount Holyoke College; Andrés R. Vindas-Meléndez, University of Kentucky; Dan Khanh (Aurora) Vo, Mount Holyoke College.

We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.

A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

The topic of parking functions has wide applications in probability, combinatorics, group theory, and computer science. One generalization of the classic parking function is the interval parking function, where each car has an interval rather than a single spot of preference. We classify features of interval parking functions, and in particular show that the pseudoreachability order on interval parking functions coincides with the bubble-sort order on the symmetric group. 

Based on joint work with Emma Colaric, Ryan DeMuse, and Jeremy Martin.

In this talk, we consider inductive tools for graphs (and matroids) that preserve a kind of robustness, called connectivity.  In 1966, Tutte proved that every 3-connected graph (or matroid) other than a wheel (or whirl) has a single-edge deletion or contraction that is 3-connected.  Seymour extended this result in 1980 to show that, in addition to preserving 3-connectivity, we can preserve a given substructure, namely a 3-connected minor.  We present the long-running project joint with James Oxley and Dillon Mayhew to obtain such results for graphs (and matroids) that are internally 4-connected.

Both designs and matroids are combinatorial objects that can be described as "a finite set and a family of subsets with certain nice properties". I will introduce both objects and explain an old result that tells how to make new designs form a given design by interpreting it as a matroid. The goal of this talk is to give a q-analogue of this result. A q-analogue is roughly what happens when we generalize form finite sets to finite dimensional spaces. The q-analogues of both designs and matroids can thus be described as "a finite space and a family of subspaces with certain nice properties". I will fill in some more details and explain our result on how to make new q-analogues of designs from a given q-analogue of a design, by interpreting it as the q-analogue of a matroid.

This talk is based on joint work with Eimear Byrne, Michela Ceria, Sorina Ionica and Elif Sacikara.