  For a polynomial $P$ of degree $n$ and an $m$-tuple
$\Lambda=(\lambda_1,\dots,\lambda_m)$ of distinct complex numbers, the dope
matrix of $P$ with respect to $\Lambda$ is $D_P(\Lambda)=(\delta_{ij})_{i\in
[1,m],j\in[0,n]}$, where $\delta_{ij}=1$ if $P^{(j)}(\lambda_i)=0$, and
$\delta_{ij}=0$ otherwise. Our first result is a combinatorial characterization
of the $2$-row dope matrices (for all pairs $\Lambda$); using this
characterization, we solve the associated enumeration problem. We also give
upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that
the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $\Lambda$ is
maximized when $\Lambda$ is generic. Finally, we resolve an ``extension''
problem of Nathanson and present several open problems.
