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A tale of two polytopes 1: The bipermutohedron.

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.