Trees, forests, and total positivity: I. $q$-trees and $q$-forests matrices

TL;DR

This paper introduces $q$-generalizations of matrices counting forests and trees, providing combinatorial interpretations and proving their coefficientwise total positivity using planar networks and the Lindström-Gessel-Viennot lemma.

Contribution

It presents new $q$-matrix constructions for forests and trees, along with combinatorial interpretations and proofs of total positivity, extending to polynomials in multiple indeterminates.

Findings

Matrices are coefficientwise totally positive.

Combinatorial interpretations of $q$-statistics on forests and trees.

Conjectures on Hankel-total positivity of row-generating polynomials.

Abstract

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and trees on $n+1$ vertices where $k$ children of the root are smaller than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindstr\"om-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in \emph{eight} indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.