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

TL;DR

This paper investigates the face structure of flow polytopes related to the complete graph, providing formulas and generating functions for their $f$-vectors, extending known results to more general netflow vectors.

Contribution

It introduces explicit formulas and generating functions for the $f$-vector of flow polytopes of the complete graph, including general netflow vectors, building on and extending prior work.

Findings

Derived formulas for $f$-vectors of flow polytopes

Extended results to arbitrary netflow vectors

Reproduced the $f$-vector of the Tesler polytope

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 M\'esz\'aros--Morales--Rhoades (2017).