Counting dope matrices

For a polynomial P of degree n and an m-tuple Λ=(λ₁,…,λ_m) of distinct complex numbers, the dope matrix of P with respect to Λ is D_P(Λ)=(δ_ij)_{i∈[1,m],j∈[0,n]}, where δ_ij=1 if P^(j)(λ_i)=0, and δ_ij=0 otherwise. Our first result is a combinatorial characterization of the 2-row dope matrices (for all pairs Λ); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of m×(n+1) dope matrices, and we show that the number of m×(n+1) dope matrices for a fixed m-tuple Λ is maximized when Λ is generic. Finally, we resolve an “extension” problem of Nathanson and present several open problems.