Lattice paths and branched continued fractions II. Multivariate Lah polynomials and Lah symmetric functions

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 $i$ children. show that, if sequence $\phi$ is Toeplitz-totally positive, then triangular array totally positive and row-generating $L_n(\phi,y)$ coefficientwise Hankel-totally positive. Upon specialization we obtain results symmetric functions multivariate negative type. The type are also given by branched continued fraction. Our proofs use mainly method production matrices; matrix obtained bijection from to labeled partial Lukasiewicz paths. give second proof fraction using Euler--Gauss recurrence method.