On the f-vector of flow polytopes for complete graphs

The Chan-Robbins-Yuen polytope (𝐶𝑅𝑌_𝑛) of order 𝑛 is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph 𝐾_𝑛+1 with netflow (1,0,0,…,0,−1). The volume and lattice points of this polytope have been actively studied, however its face structure has not. We give generating functions and recurrences to compute the 𝑓-vector by using Hille's (2007) result bijecting faces of a flow polytope to certain graphs, as well as Andresen-Kjeldsen's (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes over the complete graph having other (non-negative) netflow vectors and begin a study of the face lattice of CRY_n.