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Many combinatorial matrices --- such as those of binomial coefficients,Stirling numbers of both kinds, and Lah numbers --- are known to be totallynon-negative, meaning that all minors (determinants of square submatrices) arenon-negative.  The examples noted above can be placed in a common framework: for each onethere 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 ofthe 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 ofFerrers boards, and of graph Stirling numbers of chordal graphs.

Total non-negativity of some combinatorial matrices