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 $(a_1,a_2,\dots )$, and a sequence $(e_1,e_2,\dots )$, 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 $1,x-a_1,\dots ,(x-a_1)\cdots (x-a_m)$.

We consider this general framework. For a non-decreasing sequence $(a_1,a_2,\dots )$ we establish necessary and sufficient conditions on the sequence $(e_1,e_2,\dots )$ for the corresponding matrix to be totally non-negative. As corollaries we obtain total non-negativity of matrices of rook numbers of Ferrers boards, and of graph Stirling numbers of chordal graphs.

已知许多组合矩阵（例如，二项式系数，两种斯特林数和Lah数）都是完全非负的，这意味着所有未成年人（平方子矩阵的行列式）都是非负的。

上面提到的示例可以放在一个公共框架中：每个框架都有一个不递减的序列 $（{\mathrm{一种}}_{\mathrm{1个}}，{\mathrm{一种}}_2，\dots ）$和一个序列 $（{Ë}_{\mathrm{1个}}，{Ë}_2，\dots ）$，这样 $（米，ķ）$ 矩阵的项是多项式的系数 $（X-{\mathrm{一种}}_{\mathrm{1个}}）\cdots （X-{\mathrm{一种}}_{ķ}）$ 在扩展中 $（X-{Ë}_{\mathrm{1个}}）\cdots （X-{Ë}_{米}）$ 作为多项式的线性组合 $\mathrm{1个}，X-{\mathrm{一种}}_{\mathrm{1个}}，\dots ，（X-{\mathrm{一种}}_{\mathrm{1个}}）\cdots （X-{\mathrm{一种}}_{米}）$。

我们考虑这个通用框架。对于非递减序列$（{\mathrm{一种}}_{\mathrm{1个}}，{\mathrm{一种}}_2，\dots ）$ 我们在序列上建立了必要和充分的条件 $（{Ë}_{\mathrm{1个}}，{Ë}_2，\dots ）$相应的矩阵是完全非负的。作为推论，我们获得了Ferrers板的流向数矩阵和弦图的图斯特林数矩阵的总非负性。