Title: Rudiments of Homotopy Combinatorics Abstract: Where classical combinatorics deals with finite sets of structures, homotopy combinatorics deals with finite homotopy types of structures. The basic notion is that of homotopical species. I will explain two theorems: the first (joint work with David Gepner) is a 'Joyal theorem' for homotopical species, characterising their associated analytic functors in terms of exactness conditions. The second (joint work with Imma Galvez and Andy Tonks), is a 'Schmitt theorem', to the effect that restriction species give rise to incidence coalgebras, via the notion of decomposition space. Motivation comes on one hand from program semantics (generic datatypes), and on the other hand from quantum field theory (Feynman graphs). In both cases the need for a homotopical setting comes from the presence of symmetries. (1-groupoids would actually be enough to deal with these examples, but it is practical to develop the theory in the setting of infinity-groupoids.)