Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs

TL;DR

This paper proves total positivity properties of matrices counting rooted trees and forests, generalizes these results with weighted polynomials, and uses production matrices and Riordan arrays for proofs.

Contribution

It establishes total positivity of combinatorial matrices related to rooted trees and forests, and extends these results to weighted polynomial families with Toeplitz-totally positive sequences.

Findings

The matrix with entries t_{n,k} = binom(n,k) n^{n-k} is totally positive.

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

Generalizations to weighted polynomials preserve total positivity under Toeplitz-totally positive weights.

Abstract

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\mathbf{\phi})$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, \phi_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbf{\phi}$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.