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

## TL;DR

This paper introduces multivariate Lah polynomials and Lah symmetric functions, establishing their total positivity properties and providing continued fraction representations, with proofs based on production matrices and bijections.

## Contribution

It defines generic Lah polynomials with weight sequences, proves their total positivity under Toeplitz conditions, and connects them to Lah symmetric functions and continued fractions.

## Key findings

- Triangular array of Lah polynomials is totally positive.
- Row-generating polynomials are Hankel-totally positive.
- Multivariate Lah polynomials have branched continued fraction representations.

## Abstract

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.