  Many combinatorial matrices --- such as those of binomial coefficients,
Stirling numbers of both kinds, and Lah numbers --- are known to be totally
non-negative, meaning that all minors (determinants of square submatrices) are
non-negative.
  The examples noted above can be placed in a common framework: for each one
there is a non-decreasing sequence $(a_1, a_2, \ldots)$, and a sequence $(e_1,
e_2, \ldots)$, such that the $(m,k)$-entry of the matrix is the coefficient of
the polynomial $(x-a_1)\cdots(x-a_k)$ in the expansion of
$(x-e_1)\cdots(x-e_m)$ as a linear combination of the polynomials $1, x-a_1,
\ldots, (x-a_1)\cdots(x-a_m)$.
  We consider this general framework. For a non-decreasing sequence $(a_1, a_2,
\ldots)$ we establish necessary and sufficient conditions on the sequence
$(e_1, e_2, \ldots)$ for the corresponding matrix to be totally non-negative.
As corollaries we obtain totally non-negativity of matrices of rook numbers of
Ferrers boards, and of graph Stirling numbers of chordal graphs.
