# <   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

## Federico Ardila

In our work on the Lagrangian geometry of matroids, we introduced the conormal fan of a matroid $$M$$. We used it to reinterpret the Chern-Schwartz-MacPherson cycle of $$M$$ geometrically, and to prove Brylawski and Dawson's conjectures on the log-concavity of the $$h$$-vector of $$M$$. Two related polytopes arose in our investigation of conormal fans: the bipermutohedron and the harmonic polytope. This talk will discuss the combinatorics of the bipermutohedron, a $$(2n-2)$$-dimensional polytope with $$(2n)!/2^n$$ vertices, $$3^n−3$$ facets, and an elegant face structure. In particular we compute its $$h$$-polynomial, which we call the biEulerian polynomial, and we prove that it is real-rooted. This talk will present joint work with Graham Denham and June Huh. I will try to make it as self-contained as possible. The two parts of this tale of two polytopes can be followed independently.