On the $f$-vectors of flow polytopes for the complete graph

The Chan-Robbins-Yuen polytope ($CR{Y}_n$) 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 $(1,0,0,\dots ,0,-1)$. The volume and lattice points of this polytope have been actively studied, however its face structure has received less attention. We give generating functions and explicit formulas for computing the $f$-vector by using Hille’s (2003) 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 of the complete graph having arbitrary (non-negative) netflow vectors and recover the $f$-vector of the Tesler polytope of Mészáros–Morales–Rhoades (2017).