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

• September 13 - SPECIAL SEMINAR - In special room: 317

  Speaker: Jonathan Touboul
  Title: Complex oscillations and synchrony in models of large-scale random neural networks
  Abstract: Unprecedented biological advances in neuroscience and fine descriptions of the brain based on novel data acquisition techniques have brought to light the extraordinary complex functioning of the nervous system whose dynamical behavior emerges from structured networks of interacting cells in random interactions and subject to noise. I will present two or three recent progress in the mathematical characterization of such systems allowing finer understanding of brain’s dynamics. First, using large-deviations techniques and probability theories, I will show that depending on the level of fluctuations of the interaction amplitudes between cells (synaptic weights), two distinct and somewhat universal limits describe the dynamics of randomly connected networks as the network size diverges. These are complex implicit equations in the space of stochastic processes. I will show how dynamical systems and spectral theory of operators allow describing their complex dynamics, and understanding the role of connectivity structure, delays and randomness. I will in particular exhibit a very surprising and somewhat paradoxical role of noise and heterogeneity, having the ability to synchronize neuronal activity. These are only first steps lifting a small part of the veil on a wild and largely unexplored territory at the crossroads of probability theory, random matrices, dynamical systems and geometry.
  
  
  
  

• September 20

  Speaker: Sam Hopkins (MIT)
  Title: The coincidental down-degree expectations (CDE) property for posets
  Abstract: Reiner, Tenner, and Yong recently introduced the coincidental down-degree expectations (CDE) property for posets. A poset P is CDE if the expected down-degree is the same for the uniform distribution on P and for the distribution on P that weights each element proportional to the number of maximal chains passing through that element. Reiner, Tenner, and Yong showed that many posets familiar to algebraic combinatorialists, such as certain intervals of the weak Bruhat order, are CDE. In earlier independent work with Chan, Haddadan, and Moci (which was motivated by some surprising combinational results arising from a new approach to Brill-Noether theory) we found a large family of CDE intervals of Young’s lattice. In this more recent work I extend these results to the shifted setting. These shifted results complete the case-by-case proof that all minuscule lattices are CDE, as was conjectured by Reiner-Tenner-Yong. Time permitting I will explain some applications of the study of CDE posets: product formulas for certain kinds of set-valued tableaux (i.e., product formulas for certain square-free coefficients of stable Grothendieck polynomials), and homomesy results (in the sense of Propp and Roby) for rowmotion and gyration acting on sets of order ideals.
  
  
  
  

• September 27

  Speaker: Kabir Ramola (Brandeis)
  Title: Disordered Networks and Jamming Transitions in Amorphous Media
  Abstract: Why are amorphous materials (such as sand) able to flow like liquids, while also being able to sustain stresses like solids? Several puzzling characteristics of such granular media are the subject of active research. One such phenomenon is that of 'jamming' whereby a collection of macroscopic particles acquires global rigidity with external compression. In such cases contact forces between the individual particles play an important role in the transmission of stress through the system. System spanning contact networks determine the response of such materials to external perturbations. These networks are inherently 'disordered' as the constituent particles arrange themselves into random spatial patterns. In this talk I will explain how graph theoretical techniques applied to these networks can be used to develop predictive frameworks for granular materials.
  
  
  
  

• October 11

  Speaker: Thomas McConville (MIT)
  Title: Zonotopes whose cellular strings are all coherent
  Abstract: A monotone path on a polytope is a maximal path along the edges of the polytope in some fixed direction. For example, the monotone paths on a simplex with n+2 vertices correspond to subsets of {1,..,n} and monotone paths on an n-dimensional cube correspond to permutations of {1,...,n}. In both of these cases, the set of monotone paths naturally index the vertices of some other polytope, a cube and a permutahedron respectively. In joint work with Rob Edman and Pakawut Jiradilok, we discovered some other polytopes that have a polytope of monotone paths and obtained a partial characterization of zonotopes with this property of low corank.
  
  
  
  

• October 18

  Speaker: Gabor Lippner (Northeastern University)
  Title: Measurable graph theory
  Abstract: "Graphings" are a natural generalization of finite graphs on probability measure spaces. These objects arise naturally as limits of finite graphs, as well as from the study of invariant random processes on discrete groups.
  Measurable graph theory studies graphings from the perspective of classical graph theory. It lies at the crossroads of ergodic theory and discrete mathematics. I will explain how to generalize standard notions (matchings, chromatic number, expansion, etc) to graphings and survey recent results on the surprising behavior of these notions in the measurable setting.
  
  
  
  

• November 1

  Speaker: Ira Gessel (Brandeis)
  Title: An Introduction to Symmetric Functions
  Abstract: This talk will be an introduction to the theory of symmetric functions, including the important bases for symmetric functions, the scalar product, connections with representations of symmetric groups, and plethysm.
  Slides: Slides.
  
  
  
  

• November 8

  Speaker: Ira Gessel (Brandeis)
  Title: An Introduction to Combinatorial Species
  Abstract:The theory of combinatorial species is a method for counting labeled and unlabeled structures such as graphs of various types. In particular it provides a powerful alternative approach to problems traditionally solved by Pólya theory. I will discuss the basic definitions and examples, and describe the fundamental operations on species. I will also discuss the three fundamental generating functions associated with a species, in particular, the cycle index, which is a symmetric function associated with an action of the symmetric group.
  Slides: Slides.
  
  
  
  

• November 15

  Speaker: Ira Gessel (Brandeis)
  Title: The Konvalinka-Amdeberhan conjecture and plethystic inverses
  Abstract: Motivated by Billey, Konvalinka, and Matsen's formula for unlabeled tanglegrams, Tewodros Amdeberhan and Matjaž Konvalinka independently conjectured that a generalization of their formula always gives integers. I will prove their conjecture using the fact that the plethystic inverse of an integral symmetric function is integral, and I will discuss the conjectured Schur positivity of these symmetric functions, which are related to the Lyndon (primitive necklace) symmetric functions.
  Slides: Slides.
  
  
  
  

• November 22

  Speaker: Konstantin Matveev (Brandeis)
  Title: From symmetric functions to measures on semistandard Young tableaux.
  Abstract: A simple construction allows us to move from the certain special base of the ring of symmetric functions, Macdonald polynomials, to certain measures on semistandard Young Tableaux. The study of algebraic, combinatorial and probabilistic aspects of this construction leads to many interesting questions, and we will talk about random polymers, interacting particle systems, six vertex models, and plactic algebra.
  
  
  
  

• November 29

  Speaker: Justin Troyka (Dartmouth)
  Title: Exact and asymptotic enumeration of cycles according to descent set
  Abstract: The descent set of a permutation p is the set of positions i where p(i) > p(i+1). Enumerative problems concerning descents have been studied for over a century, beginning with the classic work of MacMahon (1916). More recent is the problem of counting permutations with a given descent set and a given cycle type. Gessel and Reutenauer (1993) express this number as an inner product of symmetric functions, and they prove their formula using a bijection involving multisets of necklaces. Studying the special case of a single cycle, we use Gessel and Reutenauer's result to express the number of cycles with a given descent set as a sum of ordinary descent numbers. As a special case we get a theorem of Stanley (2007) that expresses the number of alternating cycles as a sum of Euler numbers (the number of alternating permutations). Our result has an interesting consequence: just as 1/n of all permutations of n are cycles, asymptotically 1/n of the permutations of n with a given descent set are cycles, for "almost all" descent sets. Specifically, this holds if the descent set in question has enough places where it alternates between descent and ascent, and it holds for any periodic descent set.
  
  
  
  

• December 6

  Speaker: Tim Dwyer (Dartmouth)
  Title: c-Wilf equivalences of non-overlapping permutations, the cluster method, and linear extensions of posets
  Abstract: We will discuss applications of the cluster method of Goulden and Jackson to counting occurrences of a consecutive pattern π in permutations σ, that is substrings of σ that are order isomorphic to π. The cluster method has been used previously to find many examples of c-Wilf equivalences of patterns as well as a very general sufficient condition. We use the relationship between the cluster method and linear extensions of posets to prove a necessary condition for two non-overlapping permutations to be c-Wilf equivalent.