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

Abstract

We study three combinatorial models for the lower-triangular matrix with entries t n,k = 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, φ) in infinitely many indeterminates that give a weight y to each improper edge and a weight m! φ m for each vertex with m proper children. We show that if the weight sequence φ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.