On the coefficients of Tutte polynomials with one variable at 1

TL;DR

This work studies coefficients of Tutte polynomials with one variable fixed at 1 for matroids. It derives explicit coefficient formulas: for a matroid M=(X,rk) with ground set X and rank r, $[y^j]T_M(1,y)=\sum_{t=j}^{|X|-r}(-1)^{t-j}\binom{t}{j}\sigma_{r+t}(M)$ and, dually, $[x^i]T_M(x,1)=\sum_{t=i}^{r}(-1)^{t-i}\binom{t}{i}\tau_{r-t}(M)$, where $\sigma_{r+t}(M)$ counts spanning sets and $\tau_{r-t}(M)$ counts independent sets. The paper also refines these coefficients using flats, hyperplanes, circuits, cocircuits, and duality, introducing parameters $f_1(M)$, $f_2(M)$, and $d_k(M)$ to obtain matroid analogues of known graph results, including dual forms. These matroid results extend Guan (2023) and Chen–Guo (2025) from graphs to matroids and provide combinatorial proofs of their statements, resolving related open problems. The findings connect Tutte-coefficient calculations to structural features like flats, hyperplanes, and circuits, and yield graph-level corollaries via cycle matroids. overall, the work unifies interior/exterior polynomial perspectives within a purely combinatorial framework and broadens the applicability to polymatroids as well.

Abstract

Denote the Tutte polynomial of a graph $G$ and a matroid $M$ by $T_G(x,y)$ and $T_M(x,y)$ respectively. $T_G(x,1)$ and $T_G(1,y)$ were generalized to hypergraphs and further extended to integer polymatroids by Kálmán \cite{Kalman} in 2013, called interior and exterior polynomials respectively. Let $G$ be a $(k+1)$-edge connected graph of order $n$ and size $m$, and let $g=m-n+1$. Guan et al. (2023) \cite{Guan} obtained the coefficients of $T_G(1,y)$: \[[y^j]T_G(1,y)=\binom{m-j-1}{n-2} \text{ for } g-k\leq j\leq g,\] which was deduced from coefficients of the exterior polynomial of polymatroids. Recently, Chen and Guo (2025) \cite{Chen} further obtained \[[y^j]T_G(1,y)=\binom{m-j-1}{n-2}-\sum_{i=k+1}^{g-j}\binom{m-j-i-1}{n-2}|\mathcal{EC}_i(G)|\] for $g-3(k+1)/2< j\leq g$, where $\mathcal{EC}_i(G)$ denotes the set of all minimal edge cuts with $i$ edges. In this paper, for any matroid $M=(X,rk)$ we first obtain \[[y^j]T_M(1,y)=\sum_{t=j}^{|X|-r}(-1)^{t-j}\binom{t}{j}σ_{r+t}(M),\] where $σ_{r+t}(M)$ denotes the number of spanning sets with $r+t$ elements in $M$ and $r=rk(M)$. Moveover, the expression of $[x^i]T_M(x,1)$ is obtained immediately from the duality of the Tutte polynomial. As applications of our results, we generalize the two aforementioned results on graphs to the setting of matroids. This not only resolves two open problems posed by Chen and Guo in \cite{Chen} but also provides a purely combinatorial proof that is significantly simpler than their original proofs.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1
• Theorem 3.2
• proof
• Theorem 3.3