Title:  The combinatorial structure behind the free Lie algebra

Abstract:  We explore a beautiful interaction between algebra and combinatorics in the heart of the free Lie algebra on n generators: The multilinear component of the free Lie algebra Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of partitions of an n-set tensored with the sign representation. Then we can understand the algebraic object Lie(n) by applying poset theoretic techniques to the poset of partitions whose description is purely combinatorial. We will show how this relation generalizes further in order to study  free Lie algebras with multiple compatible brackets.

Title:  Cyclotomic Factors of the Descent Set Polynomial

Abstract:  The descent set polynomial is defined in terms of the descent set statistics of a permutation and was first introduced by Chebikin, Ehrenborg, Pylyavskyy, and Readdy. This polynomial was found to have many factors which are cyclotomic polynomials. In this talk, we will continue to explore why these cyclotomic factors exist, focusing on instances of the 2pth cyclotomic polynomial for a prime p.

Happ Title:  "Generalized Abel-Rothe Polynomials" 

Happ Abstract:  This sequence of polynomials is conjectured to be of a "multi" binomial type, and we will discuss how they count certain trees and generalized parking functions.

Hedmark Title:  TBA

Hedmark Abstract:  TBA

Title:  A categorification of the Stanley symmetric chromatic polynomial

Abstract:  Given a graph G with n vertices, Stanley defined a symmetric polynomial X_G(x_1, x_2, ...) so that for every positive integer k, X_G(1,..,1,0,...) = chi_G(k) is the number of proper k-colourings of G. We build a double chain complex C_*(G) of S_n-modules so that the Frobenius series Frob_G(x;q,t) of the resulting bi-graded homology H_*(G) satisfies Frob_G(x;1,1) = X_G(x_1, x_2, ...).

Title:  "I am not Lou Billera"

Abstract:  I will present the slides from a talk Lou Billera gave at the Stanley birthday conference last summer regarding the history of f-vectors.

Title:  r-Stable Hypersimplices

Abstract:  The n,k-hypersimplices are a well-studied collection of polytopes.  Inside each n,k-hypersimplex we can define a finite nesting of subpolytopes that we call the r-stable n,k-hypersimplices.  In this talk, we will define the r-stable hypersimplices and then see that they share a nice geometric relationship via a well-known regular unimodular triangulation of the n,k-hypersimplex in which they live.  Using this relationship, we will then identify some geometric and combinatorial properties of the r-stable hypersimplices.  In doing so, we will see that a number of the properties of the n,k-hypersimplex also hold for the r-stable hypersimplices within.

Title:  Hopf Lefschetz theorem for posets

Abstract:  The Hopf-Lefschetz theorem is a classical fixed point result from topology relating the Euler characteristic and the traces of certain matrices. In this talk we will prove a generalization of this theorem to order preserving maps on posets due to Baclawski and Björner. Additionally, we will prove a number of sufficient conditions on a poset P guaranteeing that all order preserving maps on P have a fixed point.

Title:  Groupoids with weak orders

Abstract:  We discuss certain groupoids equipped with partial orders satisfying properties which abstract those of weak orders of Coxeter groups (the resulting structures are not "partially ordered groupoids" in the usual sense). In particular, we describe braid presentations of the underlying groupoids and some of the very strong closure properties of these structures under natural categorical constructions. Applied even to familiar examples such as the symmetric groups, these constructions produce interesting new structures.

Title:  The complex of not 2-connected graphs

Abstract:  With every graph property that is monotone, we can associate for every positive integer n an abstract simplicial complex. The vertices of this complex are the edges of the complete graph on n nodes, and the faces are the sets of edges having this graph property. We will present and outline the proof of a result by Babson, Bjorner, Linusson, Shareshian and Welker of the homotopy type of this simplicial complex for not 2-connected graphs.

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