Discrete Geometry and Combinatorics Seminar
Recently, O. Bernardi gave an alternative definition for the Tutte polynomial of a graph, replacing the auxiliary data of "arbitrary order of the edges" with a ribbon structure and a starting edge. He then used an algorithm to traverse the edges of spanning trees and thus construct an ordering of the edges for each spanning tree. In 2013, Tamás Kálmán introduced the interior polynomial, which is a generalization to hyergraphs of the specialization $T(x,1)$ of the Tutte polynomial. We generalize Bernardi's method to hypergraphs to obtain a Bernardi type definition for the interior polynomial. It turns out that the Bernardi process gives shellable dissections of the root polytope with a very natural shelling order. Moreover, the interior polynomial can be recovered from the $h$-vector. Joint work with Tamás Kálmán.