A recursive definition for the polymatroid Tutte polynomial

Abstract

The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial ${\mathcal{T}}_P$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for ${\mathcal{T}}_P$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for ${\mathcal{T}}_P$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently.