Combinatorial Feynman integrals and Apery

The perturbative expansion of quantum field theory associates numbers (or functions) to combinatorial graphs. These Feynman integrals are often transcendental and hard to evaluate. I will review various combinatorial invariants of graphs that behave similar to these integrals. In particular, I will explain a relation between spanning tree partitions and circuit partitions. It allows for efficient counting of these partitions, producing for every graph an integer sequence that determines the Feynman integral, conjecturally through an Apery-like limit. This is joint work with Francis Brown and Karen Yeats.