Combinatorics seminar
Combinatorics Seminar  Spring 2017
Tuesday 1pm2pm in Goldsmith 226.
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 31
Speaker: Kate Moore (Dartmouth)
Title: Patterns in Dynamical Systems
Abstract: Given a map f: [0, 1] > [0, 1], we may consider finite sequences of iterates x, f(x), f(f(x)),..., f^{n1}(x). If these n values are different, we associate a permutation \pi \in S_n to this sequence by replacing the smallest value with a 1, the second smallest with a 2, and so on. If \pi arises in this way, we say that \pi is an allowed pattern of f. It is known that if f is a piecewise monotone map, then there are permutations that are never realized by f. This surprising observation gives rise an important tool for distinguishing random from deterministic time series and for estimating the complexity of a time series.
In general, determining the allowed patterns of a given family of maps is a difficult problem and the question has only been answered in a few specific cases. Building on work by Archer and Elizalde, I will present a method to determine the allowed patterns signedshifts, negative shifts and negative betashifts.

February 7
Speaker: Yan Zhuang (Brandeis)
Title: ShuffleCompatible Permutation Statistics
Abstract: We introduce the notion of a shufflecompatible permutation statistic and the shuffle algebra associated to a shufflecompatible statistic. We give a necessary and sufficient condition for the shufflecompatibility of a descent statistic, which shows that the shuffle algebra of any shufflecompatible descent statistic is isomorphic to a quotient of the algebra of quasisymmetric functions QSym. We also give a dual version of this condition which is obtained by exploiting the duality between QSym and the coalgebra of noncommutative symmetric functions Sym. These results are used to prove the shufflecompatibility of several wellknown descent statistics and to characterize their shuffle algebras. (Joint work with Ira Gessel.)

February 14
Speaker: Yan Zhuang (Brandeis)
Title: Eulerian Polynomials and Descent Statistics
Abstract: We present several new identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials as well as refinements of type B Eulerian polynomials and flag descent polynomials, extending results of Stembridge and Petersen. We also give qexponential generating functions for qanalogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. These results are proven by a general method involving noncommutative symmetric functions. Time permitting, we present a generalization of our result for the joint distribution of the peak number and descent number, which is proven using the modified FoataStrehl action and extends a result of Brändén.
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February 28
Speaker: Mark Kempton (Harvard)
Title: Quantum state transfer on graphs
Abstract: A quantum walk on a graph G describes the evolution of the quantum state of a particle on the graph, and is described by a discrete version of the Schrodinger equation involving a graph Hamiltonian on G. If u and v are two vertices of a graph, then we say that there is perfect state transfer from u to v if there is some time at which a particle starting at vertex u ends up at vertex v. Considerable research has been done in recent years on perfect state transfer in graphs, particularly in the case where the graph Hamiltonian is take to be the adjacency matrix. In addition, one can include an energy potential on the vertex set, which amounts to adding a diagonal matrix to the Hamiltonian. I will present results showing how the potential can affect whether or not a graph admits perfect state transfer. In particular, for paths of length greater than 4, there is no potential that can be chosen for which the path admits perfects state transfer. However, there are infinite families of a graphs where a potential does induce perfect state transfer on the graph.

March 7
Speaker: Olivier Bernardi (Brandeis)
Title: Counting lattice walks in the quarter plane
Abstract: We consider lattice walks in the quarter plane N^2, with a fixed set of steps S. Depending on the set of steps S the enumerative properties of the class of walks can vary widely. In the past decade, various sophisticated methods have been used to derive these enumerative properties, and this has led to a rich collection of attractive results dealing with the nature (algebraic, Dfinite, differentiallyalgebraic,...) of the associated generating function, depending on S.
We consider a new method inspired by Tutte's work on the counting of colored triangulations. This method brings us closer to a complete classification of the enumerative properties of these walks, and a more uniform treatment of all the cases where S is contained in {1, 0,1}^2. In particular, the infamous Gessel model, can be solved quite readily by this method.
This is joint work with Mireille BousquetMelou and Kilian Raschel.

March 21
Speaker: Everett Sullivan (Dartmouth)
Title: TBA
Abstract: TBA

March 28
Speaker: Pavel Galashin (MIT)
Title: TBA
Abstract: TBA
Here are some indications for reaching Brandeis, and the math department.
Previous semesters:
Fall 2016,
Spring 2016,
Fall 2015,
Spring 2015,
Fall 2014,
Spring 2014,
Fall 2013.