In this talk we show the classical q-binomial can be expressed more compactly as a pair of statistics on a subset of 01-permutations via major index, an instance of the cyclic sieving phenomenon related to unitary spaces is also given. We then generalize this idea to q-Stirling numbers of the second kind using restricted growth words. The resulting expressions are polynomials in q and 1 + q. We extend this enumerative result via a decomposition of a new poset whose rank generating function is the q-Stirling number Sq[n,k] which we call the Stirling poset of the second kind. This poset supports an algebraic complex and a basis for integer homology is determined. This is another instance of Hersh, Shareshian and Stanton's homological version of the Stembridge q = -1 phenomenon. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done beginning with de Médicis and Leroux's rook placement formulation. Time permitting, we will indicate a bijective argument à la Viennot showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.

In this talk, we will present results by Björner and Brenti on certain affine permutations. We show that the group of affine permutations realizes the affine Coxeter group of type A. We then consider some combinatorial properties of affine permutations. We exhibit bijections between affine inversion tables, defined analogously to the usual inversion tables of permutations, and certain integer partitions. Finally, we examine several partial orders on affine permutations.

We may describe a polytope P as the convex hull of n points in space. Knowing the numbers of chains of faces of P there are in each dimension is interesting. The toric g-vector and CD-index of P are useful invariants for encoding this information. For a simplicial polytope P, Lee defined the winding number w_k in a Gale diagram corresponding to P. He showed that w_k in the Gale diagram equals g_k of the corresponding polytope. In this talk, we will introduce the basic notions and briefly explain the simplicial case before focusing on our work in the general cases. We will show how to determine g_k of polytopes in certain cases by only considering the corresponding Gale diagram. In particular, we will use Gale diagrams to determine g-vectors of polytopes with 2-dimensional Gale diagrams. Further, we will extend the generalized lower bound theorem to nonpyramids with few vertices. Then we will discuss how to obtain the CD-index of polytopes dual to polytopes with 2-dimensional Gale diagrams.

Here I discuss an interesting polynomial called the cover polynomial which encodes information on covering directed graphs by vertex-disjoint directed paths and directed cycles. For most part of this talk, a theorem stating a relationship between the cover polynomial of a digraph and that of its dual will be discussed.

We study colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group acts on the multilinear components of these algebras and we use poset topology techniques to understand these representations. We introduce a poset of weighted subsets and prove that the multilinear components of the colored exterior algebra are isomorphic as representations to the top cohomology of its maximal intervals. We use this isomorphism and a technique of Sundaram to compute the multiplicities of the irreducibles inside these representations.

We will show how a Hopf algebra structure on graphs can be lifted to abstract simplicial complexes. We make use of this Hopf structure to study the antipode map of several families of Combinatorial Hopf algebras arising this way. We will see how these antipodes extend and recover Stanley's (-1)-color theorem, namely, the number of acyclic orientations in a graph can be obtained by evaluating its chromatic polynomial at -1. Finally, we will discuss some work in progress aiming to obtain a q-deformation of the Hopf algebras in question that would specialize to Shareshian-Wachs chromatic quasisymmetric function. This is joint work with J. Hallam, J. Machacek. We will provide the necessary background on Combinatorial Hopf algebras.

Hook Schur functions were introduced by Berele and Regev in their study of the representation theory of the general linear Lie superalgebra. These functions have representation-theoretic significance as characters of a certain S_n representation and are also interesting from a combinatorial standpoint. Quasisymmetric hook Schur functions refine the hook Schur functions in a natural way and have similar combinatorial properties. In this talk both types of functions will be introduced along with some of the main combinatorial results such as analogues of RSK and Littlewood-Richardson rules.

We discuss integrality and the integer decomposition property of Gelfand-Tsetlin polytopes. In particular, we completely characterize which GT-polytopes which are integral, in the case corresponding to standard Young tableaux. If there is time, we discuss some related counter-examples to natural questions that only appear in high dimensions.

Meander graphs, introduced by Dergachev and A. Kirillov, provide a method for determining the index of seaweed subalgebras of the special linear Lie algebras. In the case of a Frobenius seaweed, we use the meander to prove that the spectrum of the adjoint of a principal element is an unbroken sequence of integers. Additionally, we show that the sequence of the dimensions of the associated eigenspaces are symmetric. Symplectic seaweed subalgebras enjoy the same properties, and we use symplectic meanders to prove this.

If V is a finite set of points in Euclidean space and x is a point in the convex hull of V, then usually there are infinitely many choices for expressing x as a convex combination of the points in V. Which one is "best" or distinguished in some particular way? We will provide a candidate by examining the moment map associated with a particular toric variety.

The cause of unimodality for Ehrhart h* vectors of lattice polytopes remains mysterious. I will provide a brief survey of this research area, including an update on current projects regarding Ehrhart h* vectors for special families of Fano lattice simplices. This is joint work with Robert Davis and Liam Solus.

Discrete dynamical systems are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. This talk will survey the main results about nested canalizing functions and an extension of the concept into the multistate case. It will also introduce the concept of layers of canalization and its relevance in the dynamics of networks made up of nested canalizing rules. Finally, since dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming, we will discuss the role of canalization in the control of Boolean molecular networks.

The Ehrhart polynomials of integral convex polytopes count integer points under dilations of the polytopes. In this talk, I will discuss the possible sign patterns of the coefficients of Ehrhart polynomials of integral convex polytopes. While the leading terms, the second leading terms and the constant of Ehrhart polynomials are always positive, the other terms aren't necessarily positive. In fact, some examples of Ehrhart polynomials with negative coefficients were known before. For arbitrary dimension, I will describe a construction of Ehrhart polynomials with negative coefficients.