Immanant Positivity for Catalan-Stieltjes Matrices

Abstract

In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials and Narayana polynomials.

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

“…We have verified the immanant positivity of all square submatrices of H n for n ≤ 6 by Sage [21]. Note that, our method in [13] does not apply to H directly, since there exist some arcs weighted by −1 in our planar network for H.…”

“…When λ = (1 n ), the immanant Imm λ M specializes to det[M]. In [13] we proved the immanant positivity for a large family of Catalan-Stieltjes matrices and their associated Hankel matrices. This motivates us to study the the immanant positivity for the Hankel matrix H defined as in (1.1).…”

“…with b 0,0 (q) = 1 and b n,k (q) = 0 for k > n. The triangular array B belongs to a wide class of matrices, called q-recursive matrices in [25], or Catalan-Stieltjes matrices in [18,13], which we will recall below. Let γ = (r k (q)) k≥0 , σ = (s k (q)) k≥0 and τ = (t k (q)) k≥1 be three sequences of polynomials in q.…”

“…In this paper, inspired by our recent work [13], we construct for H a suitable planar network allowing negative weights and solve Pan and Zeng's problem. In our construction, the planar network for H can be naturally divided into serial segments which are essentially subnetworks of the planar network for the coefficient matrix L B of B.…”