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

Abstract

We introduce the generic Lah polynomials $L_{n,k}(\varphi )$, 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 $\varphi$ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials $L_n(\varphi ,y)$ 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 Lukasiewicz paths. We also give a second proof of the continued fraction using the Euler--Gauss recurrence method.

arXiv comment51 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1807.03271. Version 2 adds a new Section 8, using exponential Riordan arrays to give a quick alternate proof of Proposition 1.4(a); plus a few other minor changes. To be published in the European Journal of Combinatorics

References in corpus (8)

• Lagrange Inversion
• Total positivity of Riordan arrays
• Total positivity of recursive matrices
• Catalan-like numbers and Stieltjes moment sequences
• A note on Stirling permutations
• On Path diagrams and Stirling permutations
• Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity
• Comparing two statistical ensembles of quadrangulations: a continued fraction approach

Cited by in corpus (6)

• Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
• A simple algorithm for expanding a power series as a continued fraction
• Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions
• Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials
• Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs
• Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths