# WINCOM VIRTUAL COLLOQUIUM

We are pleased to continue this colloquium for the 2020-2021 academic year.

Each week we will be hosting a Zoom seminar by members of this network. If you would like to sign up for the email reminders for this colloquium, please join our email list below. The goal of the colloquium is to encourage community and collaboration within the network and the entire combinatorics community.

You can join this meeting via the Zoom link that will be sent out weekly. You can also find information about our seminar, and many other virtual talks at Combinatorics Lectures Online.

Slides from the talks are available where possible and can be found by clicking on the title of each talk.

## SCHEDULE AND SPEAKER LIST

### DR. PAMELA E. HARRIS, WILLIAMS COLLEGE

Kostant's partition function and magic multiplex juggling sequences

Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra g as a nonnegative integral linear combination of the positive roots of g. Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height. In this talk, we present a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. This is joint work with Carolina Benedetti, Christopher R. H. Hanusa, Alejandro Morales, and Anthony Simpson.

### UPCOMING TALKS

March 4th: Dr. Christina Boucher, Unvesrity of Florida

March 11th: Dr. Emily Sergel, University of Pennsylvania

March 18th: Dr. Karen Yeates, University of WAterloo

March 25th: Dr. Karen Meagher, University of Regina

### DR. NATALIE BEHAGUE, RYERSON UNIVERSITY

Synchronizing Times for k-sets in Automata

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.

### THURSDAY DECEMBER 3RD 2PM EST

Realization Spaces of Polytopes

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.

### THURSDAY NOVEMBER 12TH 5PM EST

Affine Demazure crystals for nonsymmetric Macdonald polynomials

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.

### THURSDAY NOVEMBER 5TH 5PM EST

Counting Regular Hypergraphs

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.

### THURSDAY OCTOBER 22ND 2PM EDT

Stepping Up: New Lower Bounds on Multicolour Hypergraph Ramsey Numbers

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.

### THURSDAY OCTOBER 15TH 2PM EDT

Breaking Symmetries: Determining and Distinguishing Mycielskian Graphs

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.

### THURSDAY OCTOBER 8TH 2PM EDT

A bijection of Richard Stanley applied to certain bond lattices

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.

### THURSDAY OCTOBER 1ST 2PM EDT

Counting integer partitions with the method of maximum entropy

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.

### THURSDAY SEPTEMBER 24TH 2PM EDT

Interval parking functions

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.

### THURSDAY SEPTEMBER 10TH 12NOON EDT

Inductive tools for graphs (and matroids)

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.

### THURSDAY 3 SEPTEMBER 2PM EDT

Designs, matroids, and their q-analogues

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.

### THURSDAY 30 JULY 2PM EDT

Combinatorics of Frieze patterns

### THURSDAY 23 JULY 2PM EDT

Abstract regular polytopes for symmetric and alternating groups

### THURSDAY 16 JULY 2PM EDT

An insertion algorithm for diagram algebras

### THURSDAY 9 JULY 2PM EDT

Perfect matchings in the random bipartite geometric graph

### THURSDAY 2 JULY 2PM EDT

On the number of Integer colorings with forbidden rainbow sums

### THURSDAY 25 JUNE 2PM EDT

Strong Dimension and Threshold Strong Dimension of Graphs.

### THURSDAY 18 JUNE 2PM EDT

The Mathematician and the Mapmaker: Using Mathematics to Combat Gerrymandering

### THURSDAY 11 JUNE 2PM EDT

Total Roman Domination Edge-Supercritical and Edge-Removal-Supercritical Graphs

### THURSDAY 4 JUNE 2PM EDT

Two Geometric Problems in Extremal Topological Combinatorics

### THURSDAY 28 MAY 2PM EDT

Reconstruction from small cards

### THURSDAY 21 MAY 2PM EDT

### THURSDAY 14 MAY 2PM EDT

Dr. Fiona Skerman, Uppsala University

### THURSDAY 7 MAY 2PM EDT

### THURSDAY 30 APRIL 2PM EDT

Extremal problems of long cycles in random graphs

### THURSDAY 23 APRIL 2PM EDT

### THURSDAY 16 APRIL 3PM EDT

The midrange crossing constant and some of its uses