| This thesis demonstrates a connection between formal group laws and chromatic symmetric functions of hypergraphs, two seemingly unrelated topics in the theory of symmetric functions. A formal group law is a symmetric function of the form f(f-1( x1) + f-1(x 2) + ···) for a formal power series f( x) ∈ Q[[x]]. A hypergraph H is a generalized graph where edges can contain more than two vertices, and a coloring chi of H is proper if no edge of H is monochromatic under chi. The chromatic symmetric function XH of hypergraph H on a vertex set V is a sum of monomials corresponding to proper colorings of H..;Our main result gives a combinatorial interpretation for certain formal group laws. In particular, we show that if f(x) is a generating function for a type of combinatorial object with a certain recursive structure then the associated formal group law is a sum of chromatic symmetric functions. For example, this occurs when f( x) is the generating function for trees, permutations, lattice paths or graphs. We develop a unifying framework by defining contractible species, classes of combinatorial objects having generating functions with this property. This will also allow us to give combinatorial interpretations to products of polynomials in some well-known families of polynomials, such as the Bell, Laguerre and Hermite polynomials. Finally, we observe that many of the hypergraphs arising from formal group laws are special hypergraphs called hypertrees, and we show that the chromatic symmetric functions of hypertrees are positive in Gessel's fundamental quasisymmetric functions when they have prime-sized edges. |
