Brandeis Combinatorics Seminar - Fall 2021

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/92650710647

Please email the organizers if you would like to know the password.

September 14:

Speaker: Changxin Ding (Brandeis)

Title: Geometric bijections between subgraphs and orientations of a graph

Abstract: Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called \emph{geometric bijections}. Recently we have extended the geometric bijections to subgraph-orientation correspondences. In this talk, I will introduce the bijections and the geometry behind them.

October 5:

Speaker: Theo Douvropoulos (UMASS Amherst)

Title: Hurwitz numbers for reflection groups.

Abstract: Originally stated in 1891 and rediscovered by Goulden and Jackson in the 90's, the Hurwitz formula for the number of minimum length, transitive factorizations of permutations in transpositions is a remarkable result in enumeration that has interactions with many areas of mathematics and has been particularly popular in recent decades. In a different direction, the general saga of Coxeter combinatorics involves imagining (and sometimes proving) generalizations of theorems that are known for the Symmetric group S_n to all reflection groups W. In the case of the Hurwitz formula, this has proven difficult so far, not least because it is hard to define a notion of "transitivity" for factorizations in reflection groups.

In joint work with Joel Lewis and Alejandro Morales, we consider "full factorizations" in reflection groups; namely, factorizations of elements in reflections such that the factors generate the full group W. We give various approaches in counting such objects and prove uniform formulas for the enumeration of the "genus-0" (or smallest length) case for the class of parabolic quasi-Coxeter elements of W. This in particular includes all elements of the symmetric group, where our formula is given in terms of the collection of relative trees on the cycles of a given permutation. An extended abstract for this work was accepted as a FPSAC talk and is available at the SLC website (also the speaker's homepage).

October 12:

Speaker: Gleb Nenashev (Brandeis University)

Title: Schubert Calculus and bosonic operators

Abstract: In this talk I will present a new point of view on Schubert polynomials via bosonic operators. In particular, we extend the definition of bosonic operators from the case of Schur polynomials to Schubert polynomials. More precisely, we work with back-stable Schubert polynomials and our operators act on the left weak Bruhat order (divided difference and Monk’s rule use the right side action on permutations in my notations). Furthermore, these operators with an extra condition give sufficiently enough linear equations for the structure of the cohomology ring of flag varieties.

This approach provide a purely combinatorial method to work with Schur and Schubert polynomials without “polynomials”. Even in the case of Schur functions it gives us a lot of simple proofs, in particular I will present an elementary proof of determinant formulas for Littlewood-Richardson coefficients.

October 19:

Speaker: Guillaume Chapuy (CNRS/Universite de Paris)

Title: Maps on non-orientable surfaces and Jack polynomials

Abstract: It is well known that maps on surfaces and their relatives (branched covers of the sphere, Hurwitz numbers) have strong connections with the symmetric group and its representations. In particular, the generating function of maps or branched covers can be written as a linear combination of products of Schur functions -- a "hypergeometric 2-Toda tau-function". In 1996, Goulden and Jackson formulated the "b-conjecture" stating that if one replaces Schur functions by Jack polynomials of parameter (1+b), then the corresponding series is b-positive, with coefficients counting, in a certain sense, maps on non-orientable surfaces. I will talk about recent progress on the subject and natural developments. This is based on joint works with Valentin Bonzom and Maciej Dołęga.

October 26:

Speaker: Ira Gessel (Brandeis University)

Title: Counting acyclic digraphs by descents

Abstract: A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments and acyclic digraphs by descents. To count strong tournaments we use Eulerian generating functions. To count acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function, and we obtain a refinement of the generating function for acyclic digraphs found by Robinson and Stanley.

This is joint work with Kassie Archer, Christina Graves, and Xuming Liang.

November 2:

Speaker: Chi Ho Yuen (University of Oslo)

Title: The Critical Group of an Adinkra

Abstract: Adinkras are decorated graphs introduced by physicists to encode supersymmetric algebras, but these graphs have shown to be related to many other areas as well. We study the Laplacian of an Adinkra as a signed graph. In particular, we determine the odd component of the critical group of an Adinkra; a novel technique is to lift the Laplacian over polynomial rings (over the integers and finite fields) and consider the parallel problems there, which might be of independent interest. This is an on-going project with Kevin Iga, Caroline Klivans, and Jordan Kostiuk.

November 9:

Speaker: Philippe Nadeau (CNRS/Universite Lyon 1)

Title: Remixed Eulerian numbers

Abstract: Remixed Eulerian numbers are polynomials in $q$ refining Postnikov's mixed Eulerian numbers, which were originally defined as mixed volumes of hypersimplices. They form a natural family which includes standard $q$-analogues of binomial coefficients, Eulerian numbers and hit polynomials. We will introduce them from several elementary points of view, and exhibit some of their properties. This is joint work with Vasu Tewari (University of Hawai'i)

November 16:

Speaker: Alex McDonough (UC Davis)

Title: A Multijection of Cokernels

Abstract: I discovered an intriguing linear algebra relationship which I call a multijection. I'd like to share a version of this result, which will be understandable to a general math audience.

I used the multijection to solve an open problem about higher-dimensional sandpile groups, but I'm also curious to find other applications. Some potentially relevant ideas are "positroids", "toric varieties", "Pontryagin duality", "Ehrhart theory", and "periodic tilings".

November 30:

Speaker: Anna Weigandt (MIT)

Title: The Castelnuovo-Mumford Regularity of Matrix Schubert Varieties

Abstract: The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for the degrees of symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials.

This is joint work with Oliver Pechenik and David Speyer.

December 7:

Speaker: GaYee Park (UMass Amherst)

Title: Minimal skew semistandard Young tableaux and the Hillman--Grassl correspondence

Abstract: Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula as a positive sum over excited diagrams of products of hook-lengths. In 2018, Morales, Pak, and Panova gave a $q$-analogue of Naruse's formula for semi-standard tableaux of skew shapes. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes gave their $q$-analogue. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called the \emph{minimal SSYT}, that are equinumerous with excited diagrams via a new description of the Hillan-Grassl bijection and have a version of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape.

This is joint work with Alejandro Morales and Greta Panova.

Links to previous semesters: 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.