Immanant Positivity for Catalan-Stieltjes Matrices

TL;DR

This paper establishes conditions under which immanants of certain matrices derived from Catalan-Stieltjes matrices are nonnegative, using recursive planar networks, and applies these results to inequalities involving various combinatorial polynomials.

Contribution

It introduces new recursive planar network constructions to determine immanant positivity for Catalan-Stieltjes matrices and their Hankel matrices, unifying inequalities for multiple combinatorial polynomials.

Findings

Immanants of submatrices are nonnegative under certain conditions.

Recursive planar networks facilitate the analysis of matrix positivity.

Unified inequalities for Eulerian, Schröder, and Narayana polynomials.

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, Schr\"{o}der polynomials and Narayana polynomials.