Brandeis Combinatorics Seminar - Fall 2022

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

September 20: (on Zoom)

Speaker: Anna Pun (CUNY-Baruch College)

Title: A raising operator formula for Macdonald polynomials

Abstract: In this talk, I will first give a very brief introduction on Macdonald polynomials and nabla operators, followed by an introduction of Catalanimal, a tool that helps us to prove the shuffle theorem under any line, the extended delta conjecture and the Loehr- Warrington conjecture. I will then focus on its variant "Macanimal" which gives us an explicit raising operator formula for the modified Macdonald polynomials. Our method just as easily yields a formula for an infinite series of $GL_l$ characters which truncates to the modified Macdonald polynomials.

This is a joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse and George Seelinger.

October 11: (In person)

Speaker: Christopher Eur (Harvard University)

Title: Tautological classes of matroids

Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce a new framework, dubbed "tautological classes of matroids," that unifies, recovers, and extends these recent developments. Time permitting, we also discuss how the framework can flexibly be adapted to settings beyond matroids. Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.

October 18: (on Zoom)

Speaker: Nathan Williams (University of Texas at Dallas)

Title: Pop, Crackle, Snap (and Pow): Some Facets of Shards

Abstract: Reading cut the hyperplanes in a real central arrangement H into pieces called shards, which reflect order-theoretic properties of the arrangement. We show that shards have a natural interpretation as certain generators of the fundamental group of the complement of the complexification of H. Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid. When H is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator Pop. In this case, we use Pop to define an embedding Crackle of Reading's shard intersection order into the pure shard monoid. When H is the reflection arrangement of a finite Coxeter group, we also define a poset embedding Snap of the shard intersection order into the positive braid monoid; in this case, our three maps are related by Snap = Crackle Pop. This is joint work with Colin Defant.

October 25: (in person)

Speaker: Yan Zhuang (Davidson College)

Title: Statistics on clusters and r-Stirling permutations

Abstract: The Goulden–Jackson cluster method, adapted to permutations by Elizalde and Noy, reduces the problem of counting permutations by occurrences of a prescribed consecutive pattern to that of counting clusters, which are special permutations with a lot of structure. Recently, the speaker proved a lifting of the cluster method to the Malvenuto–Reutenauer algebra, which specializes to Elizalde and Noy's cluster method for permutations as well as new variants which refine by additional permutation statistics, including the inverse descent number ides, the inverse peak number ipk, and the inverse left peak number ilpk. Thus, if σ is a consecutive pattern and we can count σ-clusters by ides, ipk, or ilpk, then we can count all permutations by occurrences of σ jointly with the corresponding statistic.

This talk will survey results on counting clusters by the statistics ides, ipk, and ilpk. Special attention will be given to the pattern 2134...m; in joint work with Sergi Elizalde and Justin Troyka, we show that 2134...(r+1)-clusters are equinumerous with r-Stirling permutations introduced by Gessel and Stanley, and we establish some joint equidistributions between these two families of permutations.

November 1: (on Zoom)

Speaker: Vincent Pilaud (CNRS & École Polytechnique)

Title: Acyclic reorientation lattices and their lattice quotients.

Abstract: Given a directed acyclic graph D, we consider the set of its acyclic reorientations, ordered by inclusion of their sets of reversed arcs. We obtain for example a boolean lattice when D is a forest, and the weak order when D is a tournament. We will characterize the directed acyclic graphs D for which this order is a lattice, and even a semidistributive lattice. In the latter case, we will present a combinatorial model for the join irreducibles of this lattice, and show how to read the canonical join representations and the forcing order on the join irreducibles to manipulate the congruences and quotients of these lattices. This leads naturally to graphical generalizations of associahedra, permutarbrahedra, and more generally of quotientopes, constructed from graphical shards polytopes. This talk is based on http://arxiv.org/abs/2111.12387.

November 8: (in person)

Speaker: Gleb Nenashev (Brandeis)

Title: Algebras Generated By The Bott-Chern Forms On Flag Varieties And Graphs

Abstract: I will introduce the algebra generated by the Bott-Chern forms of an arbitrary flag variety. For the case of complete flag varieties these algebras were introduced by V.I.Arnold and later they were generalized to the algebras which are now known as graphical Zonotopal algebras. Zonotopal algebras are well-studied algebras, whose Hilbert series are specializations of the corresponding Tutte polynomial. I will also define a promising family of algebras that generalizes both classes. These algebras are indexed by pairs, a flag variety and a graph. Some open questions will be presented.

November 15: (on Zoom)

Speaker: Ilse Fischer (Universität Wien)

Title: Alternating Sign Matrices and Plane Partitions: A linear number of equivalent parameters

Abstract: For about 40 years now, it is known that there is the same number of alternating sign matrices (ASMs) as there is of totally symmetric self-complementary plane partitions and of descending plane partitions (DPPs). Very recently, Ayyer, Behrend and myself introduced alternating sign triangles and showed that they are also counted by the same formula. Apart from a very complicated bijective proof for the equinumerosity of ASMs and DPPs by Konvalinka and myself, explicit bijective proofs of these relations are still to be found. The discovery of equivalent parameters on two classes of objects can help in this task. We will present recent work, where we have established the step from a constant number of equivalent parameters to a linear number of such parameters. This is joint work with Hans Höngesberg and Florian Schreier-Aigner.

November 29: (on Zoom)

Speaker: Changxin Ding (Georgia Tech)

Title: Lawrence polytopes and some invariants of a graph

Abstract: We make use of two dual Lawrence polytopes $P$ and $P*$ of a graph $G$ to study the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^*$-vector of the Lawrence polytope $P$ (resp. $P^*$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^*$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope, which is obtained by dilating the polytope. By dissecting the Lawrence polytopes, we may study the $h$-vectors associated to the graph $G$ at a finer level. In particular, we understand the reduced divisors of the graph $G$ in a more geometric way.

December 6: (in person)

Speaker: Te Cao (Brandeis)

Title: Bijection and Enumeration for Type C Coxeter Arrangements

Abstract: Bijection and Enumeration for Type $C$ Coxeter Arrangements

We present bijections between regions of certain Type $C$ Coxeter arrangements and binary trees satisfying certain conditions. These bijections further allow us to obtain generating functions for the number of regions. In particular for the Type $C$ Shi arrangement, where it is well know that the number of regions is enumerated by $(2n+1)^n$ in $n$-dim, through binary trees we obtain an explicit bijection between the regions and maps from $\{1,\cdots, n\}$ to $\{-n, \cdots, n\}$. This is joint work with my advisor Professor Bernardi, generalizing his previous work in Type $A$ arrangements.

Links to previous semesters: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.