  We introduce the generic Lah polynomials $L_{n,k}(\phi)$, which enumerate
unordered forests of increasing ordered trees with a weight $\phi_i$ for each
vertex with $i$ children. We show that, if the weight sequence $\phi$ is
Toeplitz-totally positive, then the triangular array of generic Lah polynomials
is totally positive and the sequence of row-generating polynomials
$L_n(\phi,y)$ 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
Lukasiewicz paths. We also give a second proof of the continued fraction using
the Euler--Gauss recurrence method.
