Are there deletion-contraction formulas for the polymatroid Tutte polynomial?

Definitions and mathematical statement

In a common integral convention, a polymatroid on a finite ground set $E$ has a rank function $r:2^E\to{\mathbb{Z}}_{\ge0}$ with $r(\varnothing)=0$, $r(A)\le r(B)$ for $A\subseteq B$, and \[ r(A)+r(B)\ge r(A\cup B)+r(A\cap B). \] For $e\in E$, the usual rank-function deletion and contraction on $E\setminus\{e\}$ are \[ (r\setminus e)(A)=r(A),\qquad (r/e)(A)=r(A\cup\{e\})-r(\{e\}). \] The source asks whether the intended polymatroid Tutte polynomial admits deletion--contraction formulas. To turn this into a definite task, one must specify the invariant $T_r$ and the permitted form of recurrence, for example which minor polynomials, coefficients, exceptional-element cases, or additional terms may occur. The question does not assert that the matroid two-term formula holds unchanged, and no such restriction is imposed here.

The excerpt does not define "the polymatroid Tutte polynomial," its variables or normalization, or whether polymatroids are integral, real-valued, or represented by lattice bases. Several extensions of the matroid invariant are possible, so $T_r$ is not assigned an invented definition. The displayed rank and minor operations give explicit conditional context; the intended invariant and scope remain missing.

The supplied OPEN-TRIAGE summary says the Egres index labels this open, mentions related recurrences for specializations and generalizations such as greedoids and $b$-matroids, and reports that a clean general formula for a canonical polymatroid polynomial was not confirmed in that study. This is a limited supplied search assessment, not a verified nonexistence or current-open theorem.

Short English statement

Find deletion--contraction recurrences for the intended polymatroid Tutte polynomial. The source omits which polynomial, normalization, polymatroid class, and recurrence form are intended.

Sources

• Local UnsolvedMath registry, permanent ID $3000005$, exact source code catalogue entry, "Are there deletion-contraction formulas for the polymatroid Tutte polynomial?"
• Egres Open, List of open problems, page titled "Are there deletion-contraction formulas for the polymatroid Tutte polynomial?" Supplied index URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems; HTML extraction; recorded access 2026-07-29.
• No authoritative definition or research-paper citation for the intended polynomial or recurrences is supplied. The research summary reports only an unconfirmed resolution search.