The Combinatorics Seminar is an introductory seminar for combinatorics. The talk should be accesible to first year graduate students.

• September 17

  Speaker: Christian Gaetz (MIT)
  Title: A combinatorial duality and the Sperner property for the weak order.
  Abstract: A poset is Sperner if it's largest antichain is no larger than it's largest rank. In the 1980's, Stanley used the Hard-Lefschetz Theorem to prove the Sperner property for strong Bruhat orders on Weyl groups. I will describe joint work with Yibo Gao in which we prove Stanley's conjecture that the weak Bruhat order on the symmetric group is also Sperner, by exhibiting a combinatorially-defined representation of sl_2 respecting the structure of the weak and strong orders. I will explain how this representation gives rise to a combinatorial duality between the weak and strong Bruhat orders and leads to a strong order analogue of Macdonald's reduced word identity for Schubert polynomials. Time permitting, I'll also discuss some formulas for weighted chain enumeration in the strong order which generalize work of Stembridge.
  
  
  
  

• September 24

  Speaker: Cesar Cuenca (Caltech)
  Title: Probability measures and q-orthogonal polynomials
  Abstract:I will discuss joint work with Vadim Gorin and Grigori Olshanski, on remarkable families of probability measures, arising from the theory of q-hypergeometric orthogonal polynomials. These measures can be seen as sophisticated q-analogues of the well-known Plancherel measure on partitions; I will review the definition of the latter and compare it to our q-version. No knowledge of probability theory is expected.
  
  
  
  

• October 8

  Speaker: Laura Colmenarejo (UMass Amherst)
  Title: An insertion algorithm on multiset partitions with applications to diagram algebras
  Abstract: We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to new enumerative results that have representation-theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogs. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson. This is joint work with Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki
  
  
  
  

• October 22

  Speaker: Olya Mandelshtam (Brown University)
  Title: Combinatorics of the ASEP and Macdonald polynomials
  Abstract: The multispecies asymmetric simple exclusion process (ASEP) is a model of hopping particles of L different types hopping on a one-dimensional lattice of n sites. We consider the ASEP on a ring with the following dynamics: particles at adjacent sites can swap places with either rate 1 or t depending on their relative types. Recently, James Martin gave a combinatorial formula for the stationary probabilities of the ASEP with generalized multiline queues. It turns out that by introducing additional statistics on the multiline queues, we get a new formula for both symmetric Macdonald polynomials and certain nonsymmetric Macdonald polynomials. This talk is based on joint work with Sylvie Corteel (UC Berkeley) and Lauren Williams (Harvard).
  
  
  
  

• October 29

  Speaker: Pengyu Hong (Brandeis)
  Title: Spatial Pattern Modeling and Mining
  Abstract: Human cognition tends to represent complex objects/systems as hierarchical compositions of entities and their relationships. Each level of abstraction can be naturally represented as an attributed relational graph (ARG) that contains a set of vertices (representing entities) connected by edges (representing relationships between entities). The ARG representation is invariant to permutations, which is an appealing property in many applications. In this talk, I will present an approach for discovering spatial patterns embedded in multiple ARGs. A spatial pattern is modelled as a mixture of parametric graphs. We designed a method based on the expectation–maximization algorithm to estimate the maximum-likelihood parameters of a spatial pattern model from a set of ARGs. The learned model can be used to detect the instances of the spatial pattern in new samples.
  
  
  
  

• November 5

  Speaker: Kiyoshi Igusa (Brandeis)
  Title: Covering relation on the Tamari lattice.
  Abstract: This talk is about various formulas for the covering relation in the Tamari lattice starting with one given by Bernardi and Bonichan using Dyck paths and an equivalent version by Nelson using Young diagrams. I will also discuss the representation theoretic version of this story which I am much more familiar with.
  
  
  
  

• November 12

  Speaker: Changxin Ding (Brandeis)
  Title: An introduction to Hodge Theory in Combinatorics
  Abstract: In 1968, Read conjectured that for any graph, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. It was then generalized to the conjecture that the coefficients of the characteristic polynomial of any matroid form a log-concave sequence. In 2015, Adiprasito, Huh, and Katz developed the combinatorial analogues of the hard Lefschetz theorem and Hodge-Riemann relations in algebraic geometry and then resolved the general case. In this talk, I will introduce their work.
  
  
  
  

• November 19

  Speaker: Ira Gessel (Brandeis)
  Title: Counting graphs with neighborhood restrictions
  Abstract: A graph is called point-determining (or mating type) if no two vertices have the same neighborhood. An arbitrary graph can be reduced to a point-determining graph by contracting each set of vertices with the same neighborhood to a single vertex, and this decomposition enables us to give a simple generating function for counting point-determining graphs, as accomplished by Ronald Read in 1989. My former student Ji Li and I used a similar, but more complicated, decomposition to count point-determining graphs whose complements are also point-determining. In this talk I will discuss a closely related problem: counting graphs in which no two vertices have complementary neighborhoods. The decomposition approach does not work for this problem. Instead we use an approach involving inclusion-exclusion, similar to its application in rook theory, to obtain a simple generating function for these graphs. We can also count point-determining graphs without complementary neighborhoods, and several other variations.
  
  
  
  

• November 26

  Speaker: Aritro Pathak (Brandeis)
  Title: A Marstrand type Slicing theorem with mass dimension in the integers
  Abstract: The classical Marstrand projection and slicing theorems state that there is no drop in the Hausdorff dimension of a set when taking the projection in Lebesgue almost every direction in the plane; and that the intersection of the set with Lebesgue almost every line has small dimension. We talk about analogous results for the case of subsets of the plane that are 1-separated and that extend to infinity, and use the mass dimension introduced by Lima and Moreira. The projection result was proved earlier by Daniel Glasscock, and we talk about the proof of the Slicing result. As a special case of these noncompact 1-separated sets of the plane, we can talk about product sets supported on the Cartesian grid and formulate slicing statements for subsets of the integers.
  
  
  
  

• December 3

  Speaker: An Huang (Brandeis)
  Title: The Hilbert symbol as a Green's function
  Abstract: I shall explain an observation regarding the role of Tate's thesis in adelic physics, in close relation to the so-called regularized Vladimirov derivative operator in mathematical physics. In particular, the local and global functional equations for zeta integrals become the starting point of constructing a global conformal field theory on a number field with a given Hecke character. Some physics background in relation to string theory shall be briefly explained. Generalization to the quaternions shall be mentioned if time permits. This is joint work to appear with Bogdan Stoica, Shing-Tung Yau, and Xiao Zhong.