Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

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 rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered 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 Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.

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

“…Zhu [87,89] has employed closely related methods. See also Gilmore [39] for some total-positivity results for q-generalizations of tree and forest matrices, using very different methods.…”

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

“…Proof We have ψ(y, z) = a * b where a n = z n /n! and b n = y n , and hence Two interpretations of (4.26)/(2.10) in terms of digraphs are given by Gilmore [55].…”

“…Very recently, Gilmore [55] has generalized Theorem 1.1(a) to the q-forest numbers: Theorem 6.3 (Gilmore [55]). The unit-lower-triangular polynomial matrix F(q) = ( f n,k (q)) n,k≥0 is coefficientwise totally positive.…”

“…Gilmore's [55] method is very different from the one used here: he uses planar networks and the Lindström-Gessel-Viennot lemma (along the lines of [13]), not production matrices. Indeed, the production matrix…”

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

“…1 + 2q + 3q 2 + 2q 3 + q 4 1 + 2q + 2q 2 + q 3 1 0 1 + 3q + 6q 2 + 10q 3 + 12q 4 + 12q 5 + 10q 6 + 6q 7 + 3q 8 + q 9 1 + 3q + 6q 2 + 9q 3 + 10q 4 + 9q 5 + 6q 6 + 3q 7 + q 8 1 + 2q + 3q 2 + 3q 3 + 2q It follows easily from (6.18) that f n,k (q) is a monic self-reciprocal polynomial of degree (n − 1) 2 − (k − 1) 2 . Very recently, Gilmore [55] has generalized Theorem 1.1(a) to the q-forest numbers: Theorem 6.3 (Gilmore [55]). The unit-lower-triangular polynomial matrix F (q) = (f n,k (q)) n,k≥0 is coefficientwise totally positive.…”