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

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 Lindström-Gessel-Viennot lemma and establishing an involution on the set of nonintersecting families of directed paths. • Providing a combinatorial proof of the Hankel total positivity of type B Narayana polynomials. • Allowing negative weights in the construction of planar networks. • Establishing an elegant sign-reversing involution on families of nonintersecting lattice paths. [ABSTRACT FROM AUTHOR]