On the f-vector of flow polytopes for complete graphs

The Chan-Robbins-Yuen polytope (CRYn𝐶𝑅𝑌𝑛) of order n𝑛 is a
face of the Birkhoff polytope of doubly stochastic matrices that is
also a flow polytope of the directed complete graph Kn+1𝐾𝑛+1 with
netflow (1,0,0,…,0,−1)(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
f𝑓-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 CRYn𝐶𝑅𝑌𝑛.