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

• October 6

  Speaker: Ying Zhou (Brandeis)
  Title: Tamari lattices, Cambrian lattices and their application to cluster algebras
  
  
  
  

• October 20

  Speaker: Apoorva Khare (Stanford)
  Title:The critical exponent of a graph
  Abstract: Which functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history, beginning with Schoenberg and Rudin, and continues to be studied to date, for additional modern reasons. In my talk I will focus on an important special case: matrices with zeros according to a graph G, and the powers that preserve positivity when applied entrywise to all such matrices (for fixed G). Our main result classifies this set of powers for an arbitrary chordal graph G. We show how preserving positivity relates to the geometry of the graph, thus providing interesting connections between combinatorics and analysis. This yields a new graph property called the "critical exponent". We then compute the critical exponent for large families of non-chordal graphs. (Joint with D. Guillot and B. Rajaratnam.)
  
  
  

• October 27

  Speaker: An Huang (Harvard)
  Title: Graph embeddings and quadratic forms
  Abstract: What graph properties are encoded in the integral quadratic form, represented by the combinatorial Laplacian of the graph? I will present a conjectural answer to this question, obtained by considering certain graph embeddings. This is joint work in progress with S.-T. Yau and M.-H. Yueh.
  
  
  

• November 3

  Speaker:Jordan Tirrell (Brandeis)
  Title: Chain Decompositions of LYM Posets
  Abstract: Griggs conjectured in 1975 that unimodal graded posets which satisfy the LYM inequality must have a nested chain decomposition. This would generalize symmetric chain decompositions for symmetric posets. While the conjecture remains open, several partial results have been proven. In particular, we will discuss results for posets of rank 3. This is joint work from a 2007 REU with Escamilla, Nicolae, Salerno, and Shahriari. I will reflect on the progress that we made and then open up the seminar for discussion.
  
  
  

• November 17

  Speaker:Yan Zhuang (Brandeis)
  Title: Several Refinements of Pattern Avoidance and Wilf Equivalence for Permutations
  Abstract: In this talk, we will first give an expository introduction to the theory of permutation pattern avoidance—a popular subject of study in enumerative combinatorics—and the central concept of Wilf equivalence. Then, we will discuss refinements of pattern avoidance and Wilf equivalence by permutation statistics, with focus on the pioneering work of Dokos-Dwyer-Johnson-Sagan-Selsor on the major index statistic, along with preliminary results on the alternating major index by myself and Dwyer. Time permitting, we will discuss another refinement of pattern avoidance and Wilf equivalence, recently introduced by Sagan, involving quasisymmetric functions.
  
  
  

• December 8

  Speaker:Olivier Bernardi (Brandeis)
  Title: A Tutte polynomial for directed graphs,... oh wait two!
  Abstract: The Tutte polynomial is an important graph invariant. We will present two generalizations of the Tutte polynomial for directed graphs. This is joint work with Jordan Awan.