We introduce the generic Lah polynomials $L_{n,k}\left(\mathit{\varphi }\right)$, which enumerate unordered forests of increasing ordered trees with a weight ${\varphi }_i$ for each vertex with $i$ children. We show that, if the weight sequence $\mathit{\varphi }$ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials $L_n(\mathit{\varphi },y)$ is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive type are also given by a branched continued fraction. Our proofs use mainly the method of production matrices; the production matrix is obtained by a bijection from ordered forests of increasing ordered trees to labeled partial Łukasiewicz paths. We also give a second proof of the continued fraction using the Euler–Gauss recurrence method.

我们介绍通用的Lah多项式 ${\mathrm{大号}}_{ñ，ķ}（\mathit{\varphi }）$，它列举了无序森林，其中有序树木的重量增加了 ${\varphi }_{\mathrm{一世}}$ 对于每个顶点 $\mathrm{一世}$孩子们。我们证明，如果权重序列$\mathit{\varphi }$ 如果是Toeplitz完全为正，则通用Lah多项式的三角形阵列为完全正，并且行生成多项式的序列 ${\mathrm{大号}}_{ñ}（\mathit{\varphi }，ÿ）$从系数上讲汉克完全是正的。经过专门化处理，我们获得了Lah对称函数和正负类型的多元Lah多项式的结果。正型多元Lah多项式也由分支连续分数给出。我们的证明主要使用生产矩阵的方法；生产矩阵是通过从有序森林中有序增加的有序树木到标记的部分sukasiewicz路径的二射获得的。我们还使用Euler-Gauss递推法给出了连续分数的第二个证明。