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 tn,k=(nk)nn−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 tn,k(y,z) that count improper and proper edges, and further to polynomials tn,k(y,ϕ) in infinitely many indeterminates that give a weight y to each improper edge and a weight m!ϕm 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.