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

For a polymatroid P over n, Bernardi, K\'alm\'an and Postnikov Adv. Math. 402 (2022) 108355 introduced the polymatroid Tutte polynomial T₏ relying on the order 1<2<<n of n, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that T₏ 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.