On the enumeration of polymatroids

Let $p_k(n)$ be the number of $k$-polymatroids on $[n]$. We show that for every fixed $k\ge 1$, we have $$\lfloor \frac{k}{2}\rfloor \cdot \left(\genfrac{}{}{0px}{}{n}{\lfloor n\mathrm{/}2\rfloor }\right)\cdot (1+o(1))\le {\log }_2{p}_k(n)\le k\cdot \left(\genfrac{}{}{0px}{}{n}{\lfloor n\mathrm{/}2\rfloor }\right)\cdot (1+o(1))\mathrm{.}$$ We also show that for $k\ge 2$, almost all $k$-polymatroids are (i) connected, (ii) proper, and (iii) not linearly representable over any field.