John Shareshian

COMPLEXES OF NOT i-CONNECTED GRAPHS

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by Vassiliev [38, 39, 41]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n−2)! spheres of dimension 2n−5. This answers a question raised by Vassiliev in connection with knot invariants. For this case the S_n-action on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n−2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n−3)-connected graphs we provide a formula for the generating function of the Euler characteristic.

q-Eulerian polynomials: Excedance number and major index

We derive a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This arose in our work on poset topology and was presented as a conjecture at the second authors FPSAC 2006 lecture. We have since proved our conjecture and a symmetric function generalization of it. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Our proof involves an intriguing new class of symmetric functions and a bijection of Gessel and Reutenauer, which can be viewed as a necklace analog of Stanleys theory of P-partitions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Bjorner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanleys chromatic symmetric functions.

CHROMATIC QUASISYMMETRIC FUNCTIONS AND HESSENBERG VARIETIES

We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko’s representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.

On the shellability of the order complex of the subgroup lattice of a finite group

We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

Discrete Morse theory for complexes of 2-connected graphs

Abstract Using the discrete Morse theory of R. Forman, we find a basis for the unique nonzero homology group of the complex of 2-connected graphs on n vertices. This answers a question of V. Vassiliev which arises in his study of knot invariants.

Symmetric and alternating groups as monodromy groups of Riemann surfaces I : generic covers and covers with many branch points

Introduction and statement of main results Notation and basic lemmas Examples Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 Actions on

Complexes of t -Colorable Graphs

2

A new subgroup lattice characterization of finite solvable groups

-sets--the proof of Theorem 4.0.30 Actions on

Order complexes of coset posets of finite groups are not contractible

3

Topology of order complexes of intervals in subgroup lattices

-sets--the proof of Theorem 4.0.31 Nine or more branch points--the proof of Theorem 4.0.34 Actions on cosets of some