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

Abstract

For a polymatroid $P$ over $[n]$, Bernardi, Kálmán and Postnikov [\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $\mathscr{T}_{P}$ 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.

Theorems & Definitions (20)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Lemma 2.1
• proof
• Lemma 2.2