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arXiv · 2106.00656

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

Tomack Gilmore
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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.

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Tomack Gilmore. 2021-06-01. Trees, forests, and total positivity: I. $q$-trees and $q$-forests matrices. https://arxiv.org/abs/2106.00656

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