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 Shuffle-Compatibility
  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 shuffle-compatible permutation statistic and initiate an in-depth study of shuffle-compatibility, which has close connections to the theory of P-partitions, 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 shuffle-compatibility, 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 valley-hopping
  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 elementary—yet powerful and delightfully visual!—construction called "valley-hopping". My talk will be a gentle introduction to permutation enumeration through valley-hopping, and I will explain how valley-hopping 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 3-wise intersecting families
  Abstract: Ellis and Narayanan showed, verifying a conjecture of Frankl, that any 3-wise 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 3-wise 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 Non-convex Domains and Determinantal Processes
  Abstract: We show how to compute the correlation kernel for non-convex domains using skew-schur tableaux, Eynard-Mehta and other tools of the trade leading to new universal distributions.
  
  
  
  

• February 27

  Speaker: Caroline Klivans (Brown University)
  Title: On the connectivity of three-dimensional 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 Diaz-Lopez, 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 chip-firing."
  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, well-behaved 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), two-row 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 partially-ordered set (poset). It is lifted from the well-studied *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 Auslander-Reiten 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 non-intersecting 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.