Combinatorics seminar
Combinatorics Seminar  Spring 2018
When: Tuesday 3pm4pm.
Where: Gerstenzang 122.
Organizers: Olivier Bernardi and Yan Zhuang
The Combinatorics Seminar is an introductory seminar for combinatorics. The talk should be accesible to first year graduate students.

January 23 Room Gerstenzang 124
Speaker: Yan Zhuang (Brandeis)
Title: New Progress on ShuffleCompatibility
Abstract:
It has been observed since the early work of Richard Stanley that a number of permutation statistics "are compatible" with the operation of shuffling permutations. In joint work with Ira Gessel, we formalize this notion of a shufflecompatible permutation statistic and initiate an indepth study of shufflecompatibility, which has close connections to the theory of Ppartitions, quasisymmetric functions, and noncommutative symmetric functions. In this talk, I will survey the main results of our work as well as several new directions of research concerning shufflecompatibility, including preliminary work by Darij Grinberg and by Bruce Sagan.

January 30  Cancelled and replaced by a special seminar in Gerstenzang 124.
Speaker: Yan Zhuang (Brandeis University)
Title: Counting permutations using valleyhopping
Abstract:
In permutation enumeration, the area of combinatorics concerned with counting
permutations, a prototypical type of problem is to find the distribution of various
parameters called "permutation statistics" over sets of permutations. Examples of
permutation statistics include the number of "peaks" and the number of "descents",
and so we may ask: How many permutations of length n have j peaks and k
descents? Questions like this can be approached using methods from a variety of
fields, ranging from differential equations to abstract algebra, but peaks and
descents in particular can be studied using an elementaryâ€”yet powerful and
delightfully visual!â€”construction called "valleyhopping". My talk will be a gentle
introduction to permutation enumeration through valleyhopping, and I will
explain how valleyhopping arises in my own research as well as the work of my
student Richard Zhou (Lexington High School).

February 6
Speaker: Keith Frankston (Rutgers University)
Title: On regular 3wise intersecting families
Abstract:
Ellis and Narayanan showed, verifying a conjecture of Frankl, that any 3wise intersecting family of subsets of {1,2,...,n} admitting a transitive automorphism group has cardinality o(2^n), while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any 3wise intersecting family of subsets of {1,2,...,n} that is regular and increasing has cardinality o(2^n).

February 13
Speaker: Mark Adler (Brandeis University)
Title: Lozenge Tiling on Nonconvex Domains and Determinantal Processes
Abstract:
We show how to compute the correlation kernel for nonconvex domains using skewschur tableaux, EynardMehta and other tools of the trade leading to new universal distributions.

February 27
Speaker: Caroline Klivans (Brown University)
Title: On the connectivity of threedimensional tilings
Abstract:
In this talk, I will discuss domino tilings of three dimensional manifolds. In particular, I will focus on the connected components of the space of tilings of such regions under local moves. Using topological techniques we introduce two parameters of tilings: the flux and the twist. Our main result characterizes when two tilings are connected by local moves in terms of these two parameters. (I will not assume any familiarity with the theory of tilings for the talk.)

March 6
Speaker: Angelica Deibel (Brandeis University)
Title: Random Coxeter Groups
Abstract:
Coxeter groups are a class of groups which are of interest in several fields, including combinatorics and geometric group theory. In this talk, I will describe the probability model I use to study random Coxeter groups, and I will give some results.

March 13 : Talk canceled (Snow storm)

March 20
Speaker: Bruce Sagan (Michigan State University)
Title: Descent polynomials
Abstract:
A permutation \pi=\pi_1...\pi_n in the symmetric group S_n has descent set Des(\pi)={i \pi_i>\pi_{i+1}}.
Given a set I of positive integers and n>max(I), the descent polynomial of I is the cardinality d(I;n)=#{\pi in S_n  Des(\pi)=I}$.
In 1915, MacMahon proved, using the Principle of Inclusion and Exclusion, that this is a polynomial in n. Amazingly, since then properties of this polynomial do not seem to have been studied at all in the literature. We will investigate the descent polynomial in terms of its degree, coefficients when expanded in a basis of binomial coefficients, and roots.
This is joint work with Alexander DiazLopez, Pamela Harris, Erik Insko, and Mohamed Omar.

March 27
Speaker: Sam Hopkins (MIT)
Title: Ehrhart polynomial of a polytope plus dilating zonotope
Abstract: The Ehrhart polynomial of a convex lattice polytope counts the number of lattice points of that polytope as it is dilated. In 1980 Stanley proved that the coefficients of the Ehrhart polynomial of a lattice zonotope (i.e., Minkowski sum of line segments) are nonnegative integers by providing an explicit combinatorial formula. We study a kind of ``generalized Ehrhart polynomial'' which counts the number of points in the Minkowski sum of a fixed polytope plus a dilating zonotope. We offer a slight extension of Stanley's result by showing that the coefficients of this polynomial are also nonnegative integers, and we provide an explicit formula. An important zonotope is the regular permutohedron. We give a nice combinatorial formula for the coefficients of the polynomial counting the number of points in a permutohedron plus dilating regular permutohedron. This formula relies on a subtle integrality property of slices of permutohedra. Time permitting, I will explain why this formula is relevant to "root system chipfiring."
This is joint work with Alexander Postnikov.

April 10
Speaker: Sergi Elizalde (Dartmouth College)
Title: Cyclic descents of standard Young tableaux
Abstract:
Cyclic descents of a permutation were defined by Cellini, by allowing the permutation to wrap around as if the last entry was followed by the first.
A natural question is whether a similar, wellbehaved notion of cyclic descents exists for standard Young tableaux (SYT).
We conjectured that such a notion exists for SYT of any straight shape other than a hook. Recently, this conjecture has been proved by Adin, Reiner and Roichman using nonnegativity properties of Postnikov's toric Schur polynomials. Unfortunately, the proof does not provide an explicit definition of the cyclic descent set for a specific tableau.
In this talk, I will present explicit descriptions of cyclic descent sets of SYT of rectangular shape (given by Rhoades),
tworow SYT (both straight and skew), SYT consisting of a hook plus an additional cell, and certain skew shapes.
In some cases, we also describe an action on SYT that shifts the cyclic descent set.
This is joint work with Ron Adin and Yuval Roichman.

April 17
Speaker: Tom Roby (University of Connecticut)
Title: Paths to Understanding Rowmotion on a Product of Two Chains
Abstract:
Birational rowmotion is an action on the space of assignments of rational functions
to the elements of a finite partiallyordered set (poset). It is lifted from the
wellstudied *rowmotion* map on order ideals (equivariantly on antichains) of a poset
$P$, which when iterated on special posets, has unexpectedly nice properties in terms of
periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are
constant) In this context, rowmotion appears to be related to AuslanderReiten translation
on certain quivers [Yil17], and birational rowmotion to $Y$systems of type $A_m \times
A_n$ described in Zamolodchikov periodicity.
We give a formula in terms of families of nonintersecting lattice paths for iterated
actions of the birational rowmotion map on a product of two chains. This allows us to give
a much simpler direct proof of the key fact that the period of this map on a product of
chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D. Grinberg and the author), as
well as the first proof of the birational analogue of *homomesy along files* for such
posets.
This work is joint with Gregg Musiker.

April 24
Speaker: Florian Richter (Ohio State University)
Title: The Erdos sumset conjecture
Abstract: A longstanding open conjecture of Erdos states that every subset of the integers with positive density contains a sum B+C of two infinite sets B and C. I will talk about our proof of this conjecture, which uses ideas and methods coming from Ergodic Theory, including an intersectivity theorem of Bergelson, the splitting of an arbitrary function into an almost periodic and a weak mixing component, and some borrowed techniques from Beiglboeck's proof of Jin's theorem. Joint work with Joel Moreira and Donald Robertson.
Here are some indications for reaching Brandeis, and the math department.
Previous semesters:
Fall 2017,
Spring 2017,
Fall 2016,
Spring 2016,
Fall 2015,
Spring 2015,
Fall 2014,
Spring 2014,
Fall 2013.