Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The aim of this paper is to study the criteria for coefficientwise Hankel-total ...positivity of the row-generating polynomials of generalized m-Jacobi-Rogers triangles and their applications.
Using the theory of production matrices, we present the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of the output matrices of certain production matrices. In particular, we gain a criterion for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the generalized m-Jacobi-Rogers triangle. This immediately implies that the corresponding generalized m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex 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 of exponential Rirodan arrays. In addition, we also derive some results for coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes-type 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) rth-order Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total positivity of rth-order Eulerian polynomials, which in particular implies the conjecture of Sokal on the coefficientwise Hankel-total positivity of reversed 2th-order Eulerian polynomials), (4) multivariate Ward polynomials, labeled series-parallel networks and nondegenerate fanout-free functions, (5) an array from the Lambert function and a generalization of Lah numbers and associated triangles, and so on.
We study three combinatorial models for the lower-triangular matrix with entries tn,k=(nk)nn−k: two involving rooted trees on the vertex set n+1, and one involving partial functional digraphs on the ...vertex set n. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials tn,k(y,z) that count improper and proper edges, and further to polynomials tn,k(y,ϕ) in infinitely many indeterminates that give a weight y to each improper edge and a weight m!ϕm for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.
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
!
ϕ
m
for each vertex with
m
proper children. We show that if the weight sequence
ϕ
is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.
On q-Deformed Real Numbers Morier-Genoud, Sophie; Ovsienko, Valentin
Experimental mathematics,
07/2022, Volume:
31, Issue:
2
Journal Article
Peer reviewed
Open access
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "q-analogue of a real." The construction is based on the notion of q-deformed ...rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.
Given integers 1≤k1<⋯<kl≤n−1, let Flk1,…,kl;n denote the type A partial flag variety consisting of all chains of subspaces (Vk1⊂⋯⊂Vkl) inside Rn, where each Vk has dimension k. Lusztig (1994, 1998) ...introduced the totally positive part Flk1,…,kl;n>0 as the subset of partial flags which can be represented by a totally positive n×n matrix, and defined the totally nonnegative part Flk1,…,kl;n≥0 as the closure of Flk1,…,kl;n>0. On the other hand, following Postnikov (2007), we define Flk1,…,kl;nΔ>0 and Flk1,…,kl;nΔ≥0 as the subsets of Flk1,…,kl;n where all Plücker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1,…,kl are consecutive integers. We give an elementary proof of this result (including for the case of Grassmannians) based on classical results in linear algebra and the theory of total positivity. We also show that the cell decomposition of Flk1,…,kl;n≥0 coincides with its matroid decomposition if and only if k1,…,kl are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and Fl1,3;4. Finally, we determine which notions of positivity are compatible with a natural action of the cyclic group of order n that rotates the index set.
We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, ...and the cyclically symmetric amplituhedron.
This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, ...singular values and inverses of these matrices. Numerical examples are included.
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