On the coefficients of interior and exterior polynomials of polymatroids

Abstract

The Tutte polynomial is an important invariant of graphs and matroids. Chen and Guo \emph{[Adv. in Appl. Math. 166 (2025) 102868.]} proved that for a $(k+1)$-edge connected graph $G$ and for any $i$ with $0\leq i <\frac{3(k+1)}{2}$, $$[y^{g-i}]T_{G}(1,y)=\binom{|V(G)|+i-2}{i}-\sum_{j=0}^{i}\binom{|V(G)|+i-2-j}{i-j}|\mathcal{SC}_{j}(G)|,$$ where $g=|E(G)|-|V(G)|+1$, $\mathcal{SC}_{j}(G)$ is the set of all minimal edge cuts with $j$ edges, $T_{G}(x,y)$ is the Tutte polynomial of the graph $G$, and $[y^{g-i}]T_{G}(1,y)$ denotes the coefficient of $y^{g-i}$ in the polynomial $T_{G}(1,y)$. Recently, Ma, Guan and Jin \emph{[arXiv.2503.06095, 2025.]} generalized this result from graphs to matroids and obtained the dual result on coefficients of $T_M(x,1)$ of matroids $M$ at the same time. In 2013, as a generalization of $T_{G}(x,1)$ and $T_{G}(1,y)$ of graphs $G$ to hypergraphs, K\'{a}lm\'{a}n \emph{[Adv. Math. 244 (2013) 823-873.]} introduced interior and exterior polynomials for connected hypergraphs. Chen and Guo posed a problem that can one generalize these results of graphs to interior and exterior polynomials of hypergraphs? In this paper, we solve it in the affirmative by obtaining results for more general polymatroids, which include the case of hypergraphs and also generalize the results of matroids due to Ma, Guan and Jin. As an application, the sequence consisting of these coefficients on polymatroids is proven to be unimodal, while the unimodality of the whole coefficients of matroids was obtained in 2018 by Adiprasito, Huh and Katz using Hodge theory.