Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees

Abstract

We consider the lower-triangular matrix of generating polynomials that enumerate $k$-component forests of rooted trees on the vertex set $[n]$ according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight $m!{\varphi }_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\varphi$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

LaTex2e, 65 pages. arXiv admin note: text overlap with arXiv:1907.02645. Version 2 makes some minor changes. To be published in Monatshefte für Mathematik

References in corpus (6)

• Lagrange Inversion
• The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions
• A recursive algorithm for trees and forests
• A remark on the enumeration of rooted labeled trees
• Coefficientwise total positivity of some matrices defined by linear recurrences
• q-Abel polynomials

Cited by in corpus (2)

• Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions
• Exceptional sequences and rooted labeled forests