Title:  A poset view of the major index

Abstract:  We introduce the Major MacMahon map from non-commutative polynomials in the variables a and b to polynomials in q, and show how this map commutes with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset.

Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon.

Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on the signed permutations, due to Reiner.

This is joint work with Margaret Readdy

Title:  The lattice of bipartitions

Abstract:  Bipartitional relations were introduced by Foata and Zeilberger, who showed these are precisely the relations which give rise to equidistribution of the associated inversion statistic and major index. In this talk we consider the natural partial order on bipartitional relations given by inclusion, and prove that the Möbius function of each of is intervals is 0, 1, or -1. To achieve this goal we will explore the topology of the order complex. We will see that bipartitional relations on a set of size n form a graded lattice of rank 3n-2. The order complex of this lattice is homotopy equivalent to a sphere of dimension n-2. Each proper interval in this lattice has either a contractible order complex, or it is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations. The main tool in the proofs of these results is discrete Morse theory as developed by Forman, and an application of this theory to order complexes of graded posets, designed by Babson and Hersh.

This is joint work with Christian Krattenthaler.

Title:  Winding Numbers and the Generalized Lower-Bound Conjecture

Abstract:  Consider a set V of n distinct points in affinely general position in R^e. For 0 ≤ k < n−e , a k-splitter is the convex hull of a set of k points whose affine span separates the remaining points into two sets, one of which has size k. Let p be an additional point in affinely general position with respect to V . In this talk, we will discuss w_k(p), the kth winding number, which counts how many times k-splitters wrap around p in the counter clockwise direction. It is known that w_k(p) ≥ 0, as a consequence of the g-theorem. There are elementary proofs (Lee, Welzl) for some special cases (e.g. e = 2). We will discuss the ideas behind these proofs.  Then we will discuss the relationship of this work to the g-theorem.  We will then pose possible directions for further research

Title:  Examining the Ehrhart Series of Reflexive Polytopes

Abstract:  The Ehrhart series of a lattice polytope encodes combinatorial data about its integer scalings. From this series, we can determine properties of the polytope that may have been otherwise obscured, such as when a corresponding semigroup algebra is Gorenstein. In this talk, we will discuss an open question about the form of the Ehrhart series for integrally closed, reflexive polytopes and describe progress in the case of simplices.

Title:  Rademacher--Carlitz Polynomials

Abstract:  We introduce and study the Rademacher--Carlitz polynomial These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view the Rademacher—Carlitz polynomial as a polynomial analogue (in the sense of Carlitz) of the Dedekind--Rademacher sum, which appears in various number-theoretic, combinatorial, geometric, and computational contexts.  Our results come in three flavors: we prove a reciprocity theorem for Rademacher--Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms of any rational polyhedron P, and (if time allows) we derive a novel reciprocity theorem for Dedekind--Rademacher sums, which follows naturally from our setup.

This is joint work with Matthias Beck.

Title:  The number of cycles in the graph of 312-avoiding permutations

Abstract:  The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length d in the subgraph of overlapping 312-avoiding permutations.

This is joint work with Sergey Kitaev and Einar Steingrimsson.

Title:  A Lattice Path Interpretation of the Diamond Product

Abstract:  The diamond product is a poset operation that corresponds to the Cartesian product of polytopes.  The effect the diamond product has on the cd-indices of posets has been previously studied using coproducts, which resulted in recursive formulas for the product of two cd-polynomials and showed that the product is non-negative on cd-indices.  In this talk, I will give a combinatorial interpretation for the diamond product of two cd-monomials that involves weighted lattice paths.

Title:  A Little TLC

Abstract:  Every semester the research triangle in North Carolina hosts the Triangle Lectures in Combinatorics (TLC). Since not everyone was able to attend TLC this semester, we thought we would bring TLC to you.

George Andrews studies Mock-theta functions. Sarah Nelson will present highlights from Dr. Andrews' two talks, including history behind Ramanujan's Lost Notebook and some Rogers-Ramanujan identities.

Matt Beck surveys old and new theorems on graph polynomials. Florian Kohl will report some key results from this talk through the 'lens of discrete geometry' with the goal of weaving a unifying thread through some results on graph polynomials.

Victoria Powers presented on what she calls "Certificates of Positivity".  Lola Davidson and Wesley Hough will cover some basic theory and practice of these Certificates.

Robin Pemantle presented on a new lattice recursion, called the Hexahedron recursion, and some of its consequences.  Liam Solus will introduce the concept of a lattice recursion with some classic examples and then discuss Robin Pemantle's findings.

Yue Cai will be the MC for this event.

Title:  Classification of Ehrhart polynomials of integral simplices

Abstract:  The Ehrhart polynomials of an integral convex polytope is the counting function of the integer points contained in its dilation. In this talk, by using two well-known inequalities, we classify all the possible Ehrhart polynomials of integral convex polytopes with small normalized volumes. Moreover, we also discuss the Ehrhart polynomials of integral simplices with prime normalized volume

The Hilbert series of the Garsia-Haiman module can be defined combinatorially as generating functions of certain fillings of Ferrers diagrams. One of the challenges in working with the combinatorial definition is the large number of fillings needed to generate a polynomial. In this talk we look at combinatorial proof of some factorizations of the Hilbert series.