A direct proof of well-definedness for the polymatroid Tutte polynomial

For a polymatroid P over $\left[n\right]$, Bernardi et al. (2022) [1] introduced the polymatroid Tutte polynomial $T_P$ relying on the order $1<2<\cdots <n$ of $\left[n\right]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $T_P$ is equivalent to another polynomial that only depends on P. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.