Abstract: We will explore connections between model polytopes, arising in the study of log-linear models, and enumerative combinatorics. In particular, we focus on the class of decomposable models, which exhibits rich combinatorial structure and includes models related to multiset permutations. A key property of decomposable models that enables this interaction is the existence of a unimodular triangulation, shown by Hosten and Sullivant. In special cases, we show that the facets of this triangulation have a meaning in terms of descents of multiset permutations. We will see some results and conjectures in this "new" direction, based on ongoing work with Liam Solus.