  The Tutte polynomial is a crucial invariant of matroids. The polymatroid
Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi et al., 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 monotonicity properties, for
containment and for minors of the interior polynomial
$x^{n}\mathscr{T}_{P}(x^{-1},1)$ and the exterior polynomial
$y^{n}\mathscr{T}_{P}(1,y^{-1})$, for polymatroids $P$ over $[n]$. We show by a
counter-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, for
all $k\geq 0$, we characterize hypergraphs $\mathcal{H}=(V,E)$ so that the
coefficient of $y^{k}$ in the exterior polynomial of the associated polymatroid
$P_{\mathcal{H}}$ attains its maximal value $\binom{|V|+k-2}{k}$.
