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

Sokal, Alan D.

doi:10.48550/arxiv.2105.05583

Cited by 2 publications

(24 citation statements)

References 78 publications

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“…More generally, the convolution z n = k≥0 f n+1,k+1 x k y n−k preserves Stieltjes moment property of sequences. Some of results above were also independently proved in [90]. Note that T (t) = te −t , and 1…”

Section: Enumerative Labeled Trees and Forestsmentioning

confidence: 68%

“…More and more combinatorial polynomials were proved to have such properties. For example, the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Ramanujan polynomials, Dowling polynomials, Jacobi-Stirling polynomials, and so on, are q-log-convex (see Chen et al [20,21], Liu and Wang [58], Zhu [103,104,105,106], Zhu and Sun [113] for instance), 3-q-log-convex (see [107]) and q-Stieltjes moment (see [90,101,107,109]). We refer the reader to [73,74,90,91,110,111,112] for coefficientwise Hankel-total positivity in more indeterminates.…”

Section: Definitions and Notation From Total Positivitymentioning

confidence: 99%

“…which for k = 0 reduces to the row-generating polynomials M n (q) of M. Following [90], we call the matrix M(q) the binomial row-generating matrix. In fact, the definition (1.12) can be written in matrix form as…”

Section: Rogers Trianglementioning

confidence: 99%

“…(iii) The polynomial sequence (ψ Y n+1,1 ) n≥0 is coefficientwise Hankel-totally positive in q. From [90], we know that ψ Y n+1,1 = n k=1 (k − 1)!f n,k q k , which enumerates connected functional digraphs on the vertex set [n] with a weight x for each cyclic vertex; equivalently, they enumerate cyclically ordered forests of rooted trees on the vertex set [n] with a weight x for each tree. Its exponential generating function is − log(1 − qT (t)).…”

Section: Enumerative Labeled Trees and Forestsmentioning

confidence: 99%

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Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Zhu¹

2022

Preprint

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Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. Recently, based on lattice paths and branched continued fractions, Pétréolle, Sokal and Zhu (Mem. Amer. Math. Soc., to appear) developed the theory for the coefficientwise Hankel-total positivity of the m-Stieltjes-Rogers polynomial sequence in all the indeterminates. The aim of this paper is to study the criteria for the row-generating polynomial sequence of the m-Jacobi-Rogers triangle being coefficientwise Hankel-totally positive and their applications.Using the theory of production matrices, we gain a criterion for the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the m-Jacobi-Rogers triangle. This immediately implies that the corresponding m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and logconvex of higher order in all the indeterminates. In consequence, for m = 1, we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m, combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence for exponential Rirodan arrays. In addition, we also derive some results for the coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) rthorder Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total

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Section: Enumerative Labeled Trees and Forestsmentioning

confidence: 68%

Section: Definitions and Notation From Total Positivitymentioning

confidence: 99%

Section: Rogers Trianglementioning

confidence: 99%

Section: Enumerative Labeled Trees and Forestsmentioning

confidence: 99%

See 2 more Smart Citations

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Zhu¹

2022

Preprint

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“…The total positivity of the forests and trees matrices has been proven very recently in [66,67] using different methods to the ones we employ in the present paper. Sokal observes in [66] that F is the exponential Riordan array R[F, G] with F (t) := 1 and G(t) the tree function [16]…”

Section: Introductionmentioning

confidence: 94%

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

Gilmore¹

2021

Preprint

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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 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.

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

Cited by 2 publications

References 78 publications

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

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

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