Abstracts
for the Seminar

Fall 2024

**Speaker: **William Dugan, University of Massachusetts

**Title: **On the f-vector of flow polytopes for complete graphs

**Time:** 2:30 PM, Monday, November 18, 2024

**Place:** Malott 206

**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 been studied less. We give explicit formulas and generating
functions for the $f$-vector of $CRY_n$ 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 arbitrary (non-negative) netflow
vectors and study the face lattice of $CRY_n$.

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