A planar network proof for Hankel total positivity of type $B$ Narayana polynomials

TL;DR

This paper provides a planar network proof for the total positivity of type B Narayana polynomials' Hankel matrices, using a novel construction with negative weights and combinatorial involutions.

Contribution

It introduces a new planar network approach with negative weights to prove Hankel total positivity for type B Narayana polynomials, complementing existing algebraic proofs.

Findings

Established a planar network with negative weights for type B Narayana polynomials

Applied Lindström-Gessel-Viennot lemma to demonstrate total positivity

Constructed an involution on nonintersecting path families

Abstract

The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindstr\"om-Gessel-Viennot lemma and establishing an involution on the set of nonintersecting families of directed paths.