A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube and the Crapo $\beta$-invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified and condensed. Final version to appear in J. Combin. Theory Ser.

An Introduction to Combinatorics via Cayley\u27s Theorem

In this paper, we explore some of the methods that are often used to solve combinatorial problems by proving Cayley’s theorem on trees in multiple ways. The intended audience of this paper is undergraduate and graduate mathematics students with little to no experience in combinatorics. This paper could also be used as a supplementary text for an undergraduate combinatorics course

Wilsonian renormalization, differential equations and Hopf algebras

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $\phi^{3}$ diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given at the conference "Combinatorics and physics" held at Max Planck Institut fuer Mathematik in Bonn in march 2007, some misprints correcte