A unified planar network approach to total positivity of combinatorial matrices and real-rootedness of polynomials

TL;DR

This paper introduces a unified planar network framework to establish total positivity of combinatorial matrices and the real-rootedness of their generating polynomials, covering many classical combinatorial sequences.

Contribution

It provides a common sufficient condition and a construction method using planar networks for total positivity and real-rootedness in various combinatorial matrices.

Findings

Proves total positivity for Stirling, Lah, Delannoy, and derangement numbers.

Establishes real-rootedness of generating functions for these sequences.

Unifies multiple results through a single network-based approach.

Abstract

We present a common sufficient condition for the total positivity of combinatorial triangles and their reversals, as well as the real-rootedness of generating functions of the rows. The proof technique is to construct a unified planar network that represent the matrix, its reversal, and the Toeplitz matrices of rows, respectively, when selecting different sets of sources and sinks. These results can be applied to the exponential Riordan arrays, the iteration matrices and the $n$-recursive matrices. As consequences, we prove the total positivity and real-rootedness properties associated to many well-known combinatorial numbers, including the Stirling numbers of both kinds (of type A and type B), the Lah numbers, the idempotent numbers, the Delannoy numbers, and the derangement numbers of type A and type B.