Poset topology and permutation enumeration
by
Michelle Wachs
University of Miami

The field of topological combinatorics explores connections between topology and combinatorics. One of the fundamental research directions in topological combinatorics was launched by Lovász's use of the Borsuk-Ulam Theorem in proving the Kneser graph coloring conjecture. Another fundamental direction was launched by Rota's influential work on the Möbius function of a partially ordered set (poset), which led to the field of poset topology. In either case, a simplicial complex is associated with a discrete structure (a graph or a poset) and its topology is studied. I will begin this talk with a brief discussion of these branches of topological combinatorics. After that I will present joint work with John Shareshian on the topology of a certain poset introduced by Björner and Welker in their study of connections between poset topology and commutative algebra. In particular, I will describe how our study of the homology of a q-analog of the Björner-Welker poset led to the discovery of a surprising new result in permutation enumeration, namely, a q-analog of a formula of Euler for the exponential generating function of the Eulerian polynomials, involving the permutation statistics major index and excedance number. I will also briefly discuss connections between the representation of the symmetric group on the homology of the Björner-Welker poset and various results that have appeared in the literature such as MacMahon's formula for multiset derangements and work of Procesi and Stanley on toric varieties of Coxeter complexes.

Date received: February 22, 2009