Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees

TL;DR

This paper proves total positivity properties of matrices that generate polynomials counting rooted forests with various weights, extending classical results and using advanced combinatorial matrix techniques.

Contribution

It establishes total positivity of generating polynomial matrices for rooted forests and generalizes to weighted cases with Toeplitz-totally positive sequences.

Findings

Matrix of forest enumeration polynomials is coefficientwise totally positive.

Row-generating polynomials form a Hankel-totally positive sequence.

Results extend to weighted forests with Toeplitz-totally positive weights.

Abstract

We consider the lower-triangular matrix of generating polynomials that enumerate $k$-component forests of rooted trees on the vertex set $[n]$ according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight $m! \, \phi_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\phi$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.