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

Xi Chen; Alan D. Sokal

doi:10.1016/j.aam.2024.102703

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arXiv · 2302.03999

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

Xi Chen·Alan D. Sokal

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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.

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Xi Chen, Alan D. Sokal. 2023-02-08. Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs. https://doi.org/10.1016/j.aam.2024.102703

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