Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, Alan D. Sokal
We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Coefficientwise Hankel-total positivity of the Schett polynomials
本文证明了Schett多项式$X_n(x,y,z)$的偶数与奇数子序列具有系数 Hankel 全正性。
Coefficientwise Hankel-total positivity of the row-generating polynomials for the output matrices of certain production matrices
本文利用生产矩阵理论，给出了广义m-Jacobi-Rogers三角形等行生成多项式的系数式Hankel全正性判别准则，并应用于多种组合多项式，解决了Sokal提出的猜想。
We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for $m$-Dyck and $m$-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of …
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 …
Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences
本文研究了由Graham-Knuth-Patashnik递推关系定义的三角数组T(n,k;μ)，证明了其在变量参数下具有强对数凹性和强对数凸性，并展示了其Hankel-完全正性。
Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation
该研究探讨B型与移位A型Narayana三角的连续插值，证明其相关序列的全正性，且连续Narayana等变换可保持对数凸性，为组合数学相关性质提供了新的理论依据。
We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences.
Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
本文研究了枚举有根带标号树森林的生成多项式矩阵，证明了其系数全正性与Hankel全正性，并推广至通用有根森林多项式，利用生产矩阵和指数Riordan数组完成证明。
Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness
本文定义以 r≥1 参数化的高阶 Stirling 循环与子集三角，围绕四类全正性问题提出猜想，并证明 r=2 时的部分结果。
We consider real sequences $(f_n)$ that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both for the special case just mentioned and in general.
We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem} (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more…
Let $A=[a_{n,k}]_{n,k\ge 0}$ be an infinite lower triangular matrix defined by the recurrence $$a_{0,0}=1,\quad a_{n+1,k}=r_{k}a_{n,k-1}+s_{k}a_{n,k}+t_{k+1}a_{n,k+1},$$ where $a_{n,k}=0$ unless $n\ge k\ge 0$ and $r_k,s_k,t_k$ are all nonnegative. Many well-known combinatorial triangles are such matrices, including the Pascal triangle, the Stirling triangle (of the second kind), the Bell triangle, the Catalan triangles of Aigner and Shapiro. We present some sufficient conditions such that the recursive matrix $A$ is totally positive. As applications we give the total positivity of the above me…
Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs
本文研究了元素为 $t_{n,k} = \binom{n}{k} n^{n-k}$ 的下三角矩阵的组合模型，证明其全正性及行生成多项式序列的 Hankel 全正性，并利用生产矩阵与指数 Riordan 阵将其推广至无穷多不定元的多项式。
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomia…
Motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence, we define interlacing networks, which are certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to Plücker-like three-term relations among path weights. We show that indeed these relations follow from the Plücker relations in the Grassmannian together with some simple rank properties of the matrices corresponding to our interlacing networks. The space of matrice…
In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Eu…
Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes
本文通过牛顿多面体研究TSSS簇代数中F-多项式的性质，提出了簇变量Laurent表达式的递推构造，证明了多面体基的强正性，并构建了关键变量间的显式映射关系。
Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array $[\mathcal {T}_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} \mathcal {T}_{n,k}=λ(a_0n+a_1k+a_2)\mathcal {T}_{n-1,k}+(b_0n+b_1k+b_2)\mathcal {T}_{n-1,k-1}+\frac{cd}λ(n-k+1)\mathcal {T}_{n-1,k-2} \end{equation*} with $\mathcal {T}_{0,0}=1$ and $\mathcal {T}_{n,k}=0$ unless $0\le k\le n$. We derive a functional transformation for its row-generating function $\mathcal{T}_n(x)$ from the…
Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions
本文从有序双对角分解构造矩阵连分数，通过循环线性化和有理高度排序转化为块三对角形式，截断得到 Padé 型逼近，并给出收敛性、残差估计及多种谱实现。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。