On the <i>f</i> -vectors of flow polytopes for the complete graph

Abstract 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 $(1,0,0, \ldots , 0, -1)$ . 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).