The Tutte polynomial is a fundamental invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$ , introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids $P$ . In this paper, we first obtain a deletion-contraction formula for $\mathscr{T}_{P}(x,y)$ . Then we prove two natural properties of coefficientwise monotonicity, one for containment and one for minors, both for the interior polynomial $x^{n}\mathscr{T}_{P}(x^{-1},1)$ and the exterior polynomial $y^{n}\mathscr{T}_{P}(1,y^{-1})$ , where $P$ is a polymatroid over $[n]$ . We show by an example that these monotonicity properties do not extend to $\mathscr{T}_{P}(x,y)$ . Using deletion-contraction, we obtain formulas for the coefficients of terms of degree $n-1$ in $\mathscr{T}_{P}(x,y)$ . Finally, we characterize hypergraphs $=(V,E)$ such that the coefficient of $y^{k}$ in the exterior polynomial of the associated polymatroid $P_}$ attains its maximal value $\binom{|V|+k-2}{k}$ for all $k$ up to some bound.
Observations with radio arrays that target the 21-cm signal originating fromthe early Universe suffer from a variety of systematic effects. An importantclass of these are reflections and spurious couplings between antennas. Weapply a Hamiltonian Monte Carlo sampler to the modelling and mitigation ofthese systematics in simulated Hydrogen Epoch of Reionisation Array (HERA)data. This method allows us to form statistical uncertainty estimates for bothour models and the recovered visibilities, which is an important ingredient inestablishing robust upper limits on the Epoch of Reionisation (EoR) powerspectrum. In cases where the noise is large compared to the EoR signal, thisapproach can constrain the systematics well enough to mitigate them down to thenoise level for both systematics studied. Incoherently averaging the recoveredpower spectra can further reduce the noise and improve recovery. Where thenoise level is lower than the EoR, our modelling can mitigate the majority ofthe reflections and coupling with there being only a minor level of residualsystematics. Our approach performs similarly to existing filtering/fittingtechniques used in the HERA pipeline, but with the added benefit of rigorouslypropagating uncertainties. In all cases it does not significantly attenuate theunderlying signal.