Planar networks and total positivity of Riordan arrays

TL;DR

This paper provides combinatorial proofs for the total positivity of Riordan arrays using planar networks and extends some results to generate more totally positive arrays.

Contribution

It introduces combinatorial proofs via planar networks for total positivity of Riordan arrays and generalizes existing results to produce additional totally positive arrays.

Findings

Constructed planar networks with non-negative weights for proofs

Provided combinatorial proofs for several total positivity conditions

Generated new classes of totally positive Riordan arrays

Abstract

In 2015, Chen, Liang and Wang provided several sufficient conditions for the total positivity of Riordan arrays and asked for combinatorial proofs of these results. In this paper, we present such proofs by constructing suitable planar networks with non-negative weights and applying the Lindstr\"om-Gessel-Viennot lemma. Moreover, we slightly generalize one of the results and give more totally positive Riordan arrays.