Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses), constructed by Bernardi and further studied by Baker and Wang, between spanning trees and break divisors.

Revised: 2019-11-27

Accepted: 2020-05-18

Published online: 2020-10-12

Classification: 05C10, 05C31, 05C50, 05C57, 05C65

Keywords: Hypergraph, bipartite graph, ribbon structure, Tutte polynomial, interior polynomial, embedding activity, root polytope, dissection, shelling order, $h$-vector.

@article{ALCO_2020__3_5_1099_0, author = {K\'alm\'an, Tam\'as and T\'othm\'er\'esz, Lilla}, title = {Hypergraph polynomials and the Bernardi process}, journal = {Algebraic Combinatorics}, pages = {1099--1139}, publisher = {MathOA foundation}, volume = {3}, number = {5}, year = {2020}, doi = {10.5802/alco.129}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2020__3_5_1099_0/} }

Kálmán, Tamás; Tóthmérész, Lilla. Hypergraph polynomials and the Bernardi process. Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1099-1139. doi : 10.5802/alco.129. https://alco.centre-mersenne.org/item/ALCO_2020__3_5_1099_0/

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