Article

Yet another criterion for the total positivity of Riordan arrays

Authors:
Jianxi Mao at Dalian University of Technology
Jianxi Mao
Lili Mu at Jiangsu Normal University
Lili Mu
Wang Yi at Dalian University of Technology
Wang Yi
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Abstract

Let R=R(d(t),h(t)) be a Riordan array, where d(t)=∑n≥0dntn and h(t)=∑n≥0hntn. We show that if the matrix[d0h000⋯d1h1h00d2h2h1h0⋮⋮⋱] is totally positive, then so is the Riordan array R.

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... In this case, the above generating function is called a Pólya frequency formal power series. For more preliminary materials and more relevant results see, for example, Brenti [6], Mao, Mu, and Wang [17] and Pinkus [19]. Let A and B be m × m and n × n matrices, respectively. ...
... Corollary 1.3. [9,17] Let (g, f ) = (d n,k ) n,k≥0 be a Riordan array with g = n≥0 g n t n and f = n≥1 f n t n , and let (g, f ) n be the n-th order leading principle submatrix of (g, f ). Then we have the recurrence relation of (g, f ) n in the following form: ...
... is totally positive (TP), then so is R [17]. This can be proved easily by using Corollary 1.3 and mathematical induction. ...
Total Positivity of Quasi-Riordan Arrays
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Full-text available
  • Jun 2024
In this paper the total positivity of quasi-Riordan arrays is investigated with use of the sequence characterization of quasi-Riordan arrays. Due to the correlation between quasi-Riordan arrays and Riordan arrays, this study is an in-depth discussion of the total positivity of Riordan arrays.
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... where the last step follows (12), which completes the proof of (19). ...
... Mao, Mu, and Wang [19] uses (24) gives another interesting criterion for the total positivity of Riordan arrays. ...
... Denote by R r the set of all quasi-Riordan arrays defined by (19). In this section, we will show R r is a group with respect to regular matrix multiplication. ...
The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation
Preprint
  • Jun 2022
A vertical recursive relation approach to Riordan arrays is induced, while the horizontal recursive relation is represented by A- and Z-sequences. This vertical recursive approach gives a way to represent the entries of a Riordan array (g,f) in terms of a recursive linear combinations of the coefficients of g. A matrix representation of the vertical recursive relation is also given. The set of all those matrices forms a group, called the quasi-Riordan group. The extensions of the horizontal recursive relation and the vertical recursive relation in terms of c- and C- Riordan arrays are defined with illustrations by using the rook triangle and the Laguerre triangle. Those extensions represent a way to study nonlinear recursive relations of the entries of some triangular matrices from linear recursive relations of the entries of Riordan arrays. In addition, the matrix representation of the vertical recursive relation of Riordan arrays provides transforms between lower order and high order finite Riordan arrays, where the mth order Riordan array is defined by (g,f)m=(dn,k)m≥n,k≥0(g,f)_m=(d_{n,k})_{m\geq n,k\geq 0}. Furthermore, the vertical relation approach to Riordan arrays provides a unified approach to construct identities.
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... In this notation the Riordan array (g, f ) satisfies (see for instance [12]) ...
... is totally positive (TP), then so is R = (g, f ) (cf. [12] or [8,9]). Other interesting criteria for total positivity of Riordan arrays can be found in [6,7]. ...
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Full-text available
  • Jun 2024
In this paper we study the total positivity of almost-Riordan arrays (d(t)∣ g(t),f(t))(d(t)|\, g(t), f(t)) and establish its necessary conditions and sufficient conditions, particularly, for some well used formal power series d(t). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix J of an almost-Riordan array (d∣ g,f)(d|\, g,f) is presented so that J is totally positive implies the total positivity of both the almost-Riordan array (d∣ g,f)(d|\, g,f) and the Riordan array (g,f). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix J is tridiagonal, then the expressions of its principal minors are given. By using expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. A numerous examples are given to demonstrate our results.
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... is totally positive (TP), then so is R = (g, f ) (cf. [32] or [21,24]). Other interesting criteria for total positivity of Riordan arrays can be found in [10,11]. ...
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  • Nov 2024
  • ANN COMB
Let Δ\Delta be a (d−1)(d-1)-dimensional simplicial complex and hΔ=(h0Δ,…,hdΔ)h^ \Delta = (h_0^ \Delta ,\ldots , h_d^ \Delta ) its h-vector. For a face uniform subdivision operation F{\mathcal {F}}, we write ΔF\Delta _{\mathcal {F}} for the subdivided complex and HFH_{\mathcal {F}} for the matrix, such that hΔF=HFhΔh^ {\Delta _{\mathcal {F}}} = H_{\mathcal {F}}h^ \Delta . In connection with the real rootedness of symmetric decompositions, Athanasiadis and Tzanaki studied for strictly positive h-vectors the inequalities h0Δh1Δ≤h1Δhd−1Δ≤⋯≤hdΔh0Δ\frac{h_0^ \Delta }{h_1^ \Delta } \le \frac{h_1^\Delta }{h_{d-1}^ \Delta } \le \cdots \le \frac{h_d^ \Delta }{h_0^\Delta } and h1Δhd−1Δ≥⋯≥hd−2Δh2Δ≥hd−1Δh1Δ\frac{h_1^\Delta }{h_{d-1}^\Delta } \ge \cdots \ge \frac{h_{d-2}^\Delta }{h_2^\Delta } \ge \frac{h_{d-1}^\Delta }{h_1^\Delta }. In this paper, we show that if the inequalities holds for a simplicial complex Δ\Delta and HFH_{\mathcal {F}} is TP2\hbox {TP}_2 (all entries and two minors are non-negative), then the inequalities hold for ΔF\Delta _{\mathcal {F}}. We prove that if F{\mathcal {F}} is the barycentric subdivision, then HFH_{\mathcal {F}} is TP2\hbox {TP}_2. If F{\mathcal {F}} is the rth-edgewise subdivision, then work of Diaconis and Fulman shows HFH_{\mathcal {F}} is TP2\hbox {TP}_2. Indeed, in this case by work of Mao and Wang, HFH_{\mathcal {F}} is even TP.
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... This is mainly due to its occurrence in several branches of mathematics, such as classical analysis [51], representation theory [36,37,38,50], network analysis [47], cluster algebras [7,23], positive Grassmannians and integrable systems [32,33]. In combinatorics, more and more combinatorial matrices have been proved to be totally positive, such as recursive matrices [13,15], Riordan arrays [16,17,39,65], the Jacobi-Stirling triangle [41], Delannoy-like triangles [43], Catalan-Stieltjes matrices [45], Narayana triangles of types A and B [59], and the generalized Jacobi-Stirling triangle [64]. We also refer the reader to monographs [31,46] for more details. ...
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Preprint
  • Aug 2023
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... Let R L be the left product matrix of a Riordan array R. Very recently, Mao et al. [24,Theorem 3] showed that if R L is TP, then so is R. The following result can be obtained by means of the same idea used in the proof of Theorem 3.6. ...
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Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics. The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.
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Let R=(d(t),h(t)) be a Riordan array. We show that if both d(t) and h(t) are Pólya frequency formal power series, then R is totally positive.
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Here we discuss two matrix characterizations of Riordan arrays, P-matrix characterization and A-matrix characterization. P-matrix is an extension of the Stieltjes matrix defined in [28] and the production matrix defined in [8]. By modifying the marked succession rule introduced in [21], a combinatorial interpretation of the P-matrix is given. The P-matrix characterizations of some subgroups of Riordan group are presented, which are used to find some algebraic structures of the subgroups. We also give the P-matrix characterizations of the inverse of a Riordan array and the product of two Riordan arrays. A-matrix characterization is defined in [20], and it is proved to be a useful tool for a Riordan array, while, on the other side, the A-sequence characterization is very complex sometimes. By using the fundamental theorem of Riordan arrays, a method of construction of A-matrix characterizations from Riordan arrays is given. The converse process is also discussed. Several examples and applications of two matrix characterizations are presented.
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Though total positivity appears in various branches of mathematics, it is rather unfamiliar even to linear algebraists, when compared with positivity. With some unified methods we present a concise survey on totally positive matrices and related topics.
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The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems.
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Sequence characterization of Riordan arrays
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In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.
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  • Jan 1968
  • S Karlin
S. Karlin, Total Positivity, Volume 1, Stanford University Press, 1968.