Chen, Xi; Sokal, Alan D; (2024) Total positivity of some polynomial matrices that enumerate labeled trees and forests II. Rooted labeled trees and partial functional digraphs. Advances in Applied Mathematics , 157 , Article 102703. 10.1016/j.aam.2024.102703. (In press).

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Abstract

We study three combinatorial models for the lower-triangular matrix with entries tn,k = n k 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 totalpositivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

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