Brandeis Combinatorics Seminar -  Fall 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

October 20: 

Speaker: Ira Gessel (Brandeis)

Title:  Lattice paths and the Geode.

Abstract: In recent highly publicized work, Norman Wildberger and Dean Rubine studied a method for solving equations using power series. They found an interesting factorization: Let S be the power series in variables t_1, t_2, ... satisfying S = 1 + t_1 S + t_2 S^2 + .... Wildberger and Rubine showed that S-1 is divisible by t_1 + t_2 + .... They called the quotient the "Geode" and asked about its combinatorial significance. I will show that the Geode has a very simple combinatorial interpretation in terms of lattice paths.

October 27: 

Speaker: Neha Goregaokar (Brandeis)

Title:  The geometry of a counting formula for deformations of the braid arrangement 

Abstract: We consider real hyperplane arrangements whose hyperplanes are of the form $\{x_i - x_j = s\}$ for some integer $s$, which we call \textit{deformations of the braid arrangement}. In 2018, Bernardi gave a counting formula for the number of regions of any deformation of the braid arrangement $\mathcal{A}$ as a signed sum over some decorated trees. He further showed that each of these decorated trees can be associated to a region $R$ of the arrangement $\mathcal{A}$, and hence we can consider the contribution of a region to the signed sum. Bernardi showed that for \textit{transitive} arrangements, the contribution of any region of the arrangement is $1$. We remove the transitivity condition, showing that for \textit{any} deformation of the braid arrangement the contribution of a region is $1$. This is joint work with Aaron Lin. 

November 24: 

Speaker: Jonathan Fang (Brandeis)

Title:  Subgraphs Vs Orientations:infinite families of equidistributions

Abstract: A classical enumerative result states that, given a graph G and a vertex u, the number of connected subgraphs of G is equal to the number of orientations of G such that every vertex can reach u by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalence classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints. 

December 1: 

Speaker: 

Title:  

Abstract: 

December 8: 

Speaker: Alex Leighton (Brandeis)

Title:  On a Problem of Erdős: Maximal Union-Free Families of the n-Set

Abstract: 

In 1945, Paul Erdos asked for the size of the largest collection, $\mathcal{F}$, of subsets of $\{1, 2, ..., n \}$ such that for any disctinct $A,B,C \in \mathcal{F}$, $A \cup B \neq C$. He conjectured that $|\mathcal{F}| < \left( 1 + o(1)\right){n \choose n/2}$. The problem was published in a book by Ulam in 1960, where it was happened upon by Kleitman. In 1966, as a physics professor at brandeis university, Kleitman resolved Erdos' conjecture in the affirmative, proving a bound of $|\mathcal{F}| < \left( 1 + O(\sqrt{\log n / n})\right){n \choose n/2}$. Later, in 1971, Kleitman improved the bound to $|\mathcal{F}| < \left( 1 + O(1/\sqrt{n})\right){n \choose n/2}$. In 1974, Erdos and Kleitman conjectured the true bound was $|\mathcal{F}| < \left( 1 + O(1/n)\right){n \choose n/2}$. I will present on new joint work with Daniel Kleitman resolving the latter conjecture and a number of related problems in the affirmative.

Links to previous semesters: Spring2025 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.