Brandeis Combinatorics Seminar -  Spring 2025

The Brandeis Combinatorics Seminar is an introductory seminar for combinatorics.
The talks should be (at least partially) understandable to first year graduate students.

            The zoom link is  https://brandeis.zoom.us/j/94622483750

March 24: 

Speaker: Ira Gessel (Brandeis)

Title:  Congruences, continued fractions, and Smith normal forms of Hankel matrices 

Abstract: The Bell numbers, which count partitions of a finite set, are periodic modulo m for every m. This implies that there are nontrivial linear congruences that the Bell numbers satisfy. We might then ask, what is the "best" linear congruence satisfied by a given number of consecutive Bell numbers? The same question may be asked of any sequence, but we are especially interested in sequences defined by exponential generating functions, which (almost) always satisfy nontrivial congruences. In nice cases, like the Bell numbers, we can give a complete answer, which is related to orthogonal polynomials and continued fractions. In the more challenging, and therefore more interesting, cases our best tool is the Smith normal form of Hankel matrices.

March 31: 

Speaker: Neha Goregaokar (Brandeis)

Title: Interpreting the chromatic polynomial coefficients via hyperplane arrangements

Abstract:  A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements of natural unit interval graphs. We use this generalization to give a new proof of the formula for the chromatic polynomial of a natural unit interval graph.

Best, 

April 7: 

Speaker: Kiyoshi Igusa (Brandeis)

Title: From algebraic K-theory to ghost modules

Abstract: In my paper ``Generalized Grassmann invariant-redrawn'' (arXiv:2502.19147), I decided to redraw the pictures from my old paper so that they match the ``stability diagrams'' for representations of quivers which are a popular concept in representation theory. But, the diagrams don't quite match. One piece of the picture (one root of the associated Lie algebra) is missing. Using a modified version of Bridgeland stability, I will bring back the missing root as a ``ghost module''. But two ghosts appear! I will keep the exposition simple, drawing lots of pictures. I will show you how to look for ``ghosts'' with possible applications to algebraic K-theory.

April 21: 

Speaker: Grant Barkley (Harvard)

Title: The affine Tamari lattice

Abstract: The Tamari lattice is a partial order with Catalan many objects. Its elements have many incarnations, such as 312-avoiding permutations or binary trees. We introduce the cyclic and affine Tamari lattices, which are analogs of the Tamari lattice replacing the symmetric group with the affine symmetric group. The cyclic Tamari lattice is the lattice of 312-avoiding translation-invariant total orders of the integers. Its elements can also be realized as translation-invariant binary in-ordered trees. The number of elements of the cyclic Tamari lattice is a type B Catalan number. The affine Tamari lattice is a quotient of the cyclic Tamari lattice and is the first example of an extended Cambrian lattice. Its elements are in bijection with the clusters in a type D cluster algebra and its Hasse diagram is an orientation of a type D associahedron. We describe the combinatorics of these lattices and their connections with cluster algebras and the representation theory of algebras. This is joint work with Colin Defant.  

April 28: 

Speaker: Nick Ovenhouse (Yale)

Title: Mixed Dimer Models for Euler and Catalan Numbers

Abstract: The dimer model is the study of enumeration of perfect matchings of planar graphs. Mixed dimer covers are generalizations of perfect matchings, allowing the degrees to differ among the vertices. I will explain how on certain families of planar graphs, the number of mixed dimer covers is equal to the Euler numbers (which count alternating permutations) and the Catalan numbers. We can give explicit bijections, for which the standard partial order on dimer covers is mapped to the Bruhat order on associated permutations. This is joint work with Andrew Claussen.  

May 5: (in person)

Speaker: Elizabeth Bullock (Harvard)

Title: Ehrhart series of alcoved polytopes

Abstract: 

In this talk (based on joint work with Yuhan Jiang), I will describe a general method for computing the Ehrhart series of any alcoved polytope via a particular shelling order of its alcoves. As an application, we get a bijective proof of the formula for the Ehrhart h* polynomial of the second hypersimplex $\Delta_{2,n}$ in terms of Nick Early’s decorated ordered set partitions.

Links to previous semesters: Fall2024, Spring2024, Fall2023, Spring2023, Fall 2022, Spring 2022, Fall 2021, Spring 2020, Fall 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013.