Predrag Rajkovic, Marko D. Petkovic, Paul Barry
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
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Hankel determinants of Catalan-like sequences
本文计算了源于非相交Motzkin meander加权计数的类卡特兰序列的移位汉克尔行列式，其中部分为新发现或已被提出，还应用其验证了Cigler与Krattenthaler的猜想二项式行列式恒等式。
Some remarks about Hankel determinants which are related to Catalan-like numbers
该研究针对后移类Catalan数及相关序列，得出了其Hankel行列式的相关结果，属于组合数学领域的注记类研究。
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.
Shifted Hankel determinants of Catalan numbers and related results II: Backward shifts
该研究通过前置零定义序列后向移位的汉克尔行列式，得到卡特兰数及相关数与多项式序列的相关结果，并对卡特兰数卷积幂序列提出猜想。
In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation exists for more general sequences.
Some experimental observations about Hankel determinants of convolution powers of Catalan numbers
本研究通过计算机实验提出了若干关于卡特兰数卷积幂汉克尔行列式的猜想，虽多数暂无证明，但仍予以公布以期学界后续证明。
Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers
本文研究类 Catalan 数对应容许矩阵各列序列的 Hankel 行列式，归纳计算机实验揭示的性质，并通过前置零行探讨后向移位情形。
Hankel determinants of convolution powers of Catalan numbers revisited
本文利用推广的 Andrews-Wimp 定理，简洁证明了 Catalan 数卷积幂平移的 Hankel 行列式，并推广至 Narayana 多项式。
In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schröder Hankel determinants, and give a new proof of M…
This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures.
The Hankel determinants $\left(\frac{r}{2(i+j)+r}\binom{2(i+j)+r}{i+j}\right)_{0\leq i,j \leq n-1}$ of the convolution powers of Catalan numbers were considered by Cigler and by Cigler and Krattenthaler. We evaluate these determinants for $r\le 31$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin's continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjectured a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants $\left(\binom{2(i+j)+r}{i…
We consider Hankel determinants of the sequence of Catalan numbers modulo 2 (interpreted as integers 0 and 1) and more generally Hankel determinants where the sum over all permutations reduces to a single signed permutation.
Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.
Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture
本文利用反射原理、Lindström-Gessel-Viennot方法及非相交格路径的反射叠合构造，为Cigler关于卷积Catalan数Hankel行列式的猜想提供了一个双射证明。
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.
In this note we show that various natural q-analogues of the Catalan numbers can be obtained in a uniform way. Furthermore we compute their Hankel determinants.
I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the theory of orthogonal polynomials is not applicable. Even so our theorems and conjectures show some similarity with the relations between Catalan numbers and Fibonacci polynomials or between central binomial coefficients and Lucas polynomials.
We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(α, β)$ Somos $4$ sequences, $(α, 0, γ)$ Somos $6$ sequences, and $(α, β, γ, δ)$ Somos $8$ sequences, for particular values of $α$, $β$, $γ$, $δ$ which we describe. The sequences involved can be described in terms of the application of certain stretched Riordan arrays to the Catalan numbers, accompanied by a (sequence) Hankel transform. The combination of Riordan array and the Catalan numbers results from the study of certain generalized Jacobi continued fractions, based on the Counting…
Starting with a definition based on the Catalan numbers, we carry out an empirical study of the Rueppel sequence. We use the Hankel transform as the main technique. By means of this transform we find links to such sequences as the Jacobi sequence and the paper-folding sequence. We also study several number arrays defined by this sequence, which are analogs of the important Catalan triangles of combinatorics. We identify a sequence related to the Golay-Rudin-Shapiro sequence that plays a fundamental role in linking the Hankel transforms of the paper with classical sequences. Examples are given …
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures.
We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.
Dilated Hankel determinants
研究扩张的汉克尔行列式\(\ddot{H}_n\)，通过开发的六种方法，包括四种通用和两种特殊方法，对一大类序列给出简单乘积求值，并应用此解决了关于普帕尔和克鲁埃拉斯多项式根的猜想。
For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schröder paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schröder paths of length $\ell$. For constants $α, β$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\det_{1\le i,j\le n} (αr_{i+j-2}(t) +βr_{i+j-1}(t))$, $\det_{1\le i,j\le n} (αr_{i+j-1}(t) +βr_{i+j}(t))$, $\det_{1\le i,j\le n} (αs_{i+j-2}(t) +βs_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (αs_{i+j-1}(t) +βs_{i+j}(t))$ combinatorially via suitable lattice path models.
Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results
该研究将Cigler关于卡特兰数卷积幂的汉克尔行列式恒等式，推广到满足特定二次方程的一般幂级数卷积场景，并给出Motzkin数卷积幂的汉克尔行列式恒等式作为应用。
Hankel transform of a sequence obtained by series reversion II - aerating transforms
本文建立了给定整数序列的 Hankel 变换与气化变换之间的联系，并利用该联系评估了有理函数 Q(x) 的级数反演及其移位序列的 Hankel 变换。
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q…
Enumeration of consecutive patterns in flattened Catalan words
本文研究扁平化 Catalan 词上长度二和三的连续模式分布，建立含辅助参数的联合生成函数，并给出出现总数、规避者数量及同分布等价关系。
The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector $(1, 0, \ldots, 0, -1)$. The normalized volume of the Chan-Robbins-Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector $(1,1, 0, \ldots, 0, -2)$. We show that the volume of this polytope is a certain power of $2$ times the product of consecutive Catalan numbers. Our proof uses constant term identities…
The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $H_m$ and introduce accompanying noncommutative binomial coefficients.
Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers
本文研究了$q$-变形金属数的Hankel连续分数和行列式性质，证明其序列具有周期性和递推关系，并验证了相关猜想。
In this paper we show that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers are Hausdorff moment sequences in a unified approach. In particular we answer a conjecture of Liang at al. which such numbers have unique representing measures. The smallest interval including the support of representing measure is explicitly found. Subsequences of Catalan-like numbers are also considered. We provide a necessary and sufficient condition for a pattern of subsequences that if sequences are the Stielt…
In this note, we explore certain determinantal descriptions of the Robbins numbers. Techniques used for this include continued fractions, Riordan arrays and series inversion. Proven and conjectured representations involve the determinants of both Hankel and symmetric matrices. In specific cases, links are drawn to centered polygonal numbers, and to heptagons and nonagons. We conjecture a Hankel transform determinant for the Robbins numbers related to the Fibonacci and the Catalan numbers.
This work is a continuation of some recent articles presenting enumerative results for Catalan words avoiding one or a pair of consecutive or classical patterns of length $3$. More precisely, we provide systematically the bivariate generating function for the number of Catalan words avoiding a given pair of relations with respect to the length and the number of descents. We also present several constructive bijections preserving the number of descents. As a byproduct, we deduce the generating function for the total number of descents on all Catalan words of a given length and avoiding a pair o…
Efficient $k$-Sign Consistency Verification of Hankel Matrices via Schur Polynomials
通过Schur多项式高效验证Hankel矩阵的k-符号一致性，利用非负整数线性组合和Littlewood--Richardson系数证明其非负性。
In this note we use the analogy between the Catalan sequence and the Rueppel sequence to derive a variety of conjectures surrounding the Hankel transforms of a number of sequences closely related to the Rueppel sequence. Use is made of the representation of suitable generating functions by Stieltjes continued fractions. We define polynomial sequences by introducing parameters that define generalized Rueppel sequences, and we show that such polynomials have coefficient arrays that are Riordan arrays. Finally we conjecture the form of a product of Hankel transforms arising from the Rueppel seque…
The normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CRY_n$ better, the third author and Morales introduced two natural generalizations of it and conjectured that their volumes are certain powers of $2$ multiplied by a product of consecutive Catalan numbers. Zeilberger proved one of these conjectures. In this paper we present proofs of both conjectures.
Hankel Transform and $(α,β)$ Somos-4 Sequences
本文在有理函数域扩展Wang-Zhang充分条件，结合Sulanke-Xin二次变换解决Barry的(α,β)型Somos-4相关猜想，证明该序列奇偶子序列性质及汉克尔行列式相关猜想。
Bogoyavlensky lattices and generalized Catalan numbers
本文研究 Bogoyavlensky 格单位阶跃初始数据衰减问题，利用半直线终止的可线性化性质、广义 Catalan 数及其 Hankel 行列式，以及非自治对称性约化，给出精确解并推广到有限参数初值族。
Ascent sequences have received a lot of attention in recent years in connection with (2 + 2)-free posets and other combinatorial objects. Here, we first show bijectively that analogous repetition sequences are counted by the Bell numbers, and 021-avoiding repetition sequences by the Catalan numbers. Then we adapt a bijection of Chen et al and use it along with the "symbolic" method of Flajolet to find the 4-variable generating function for 021-avoiding ascent sequences by length, number of 0's, number of isolated 0's, and number of runs of 2 or more 0's. We deduce that 021-avoiding ascent sequ…
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。
In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We give closed forms for both, in terms of the original sequence, and in the case of Horadam sequences, we find the generating function of the gap-sum sequence. For some elementary sequences, we indicate that the gap-product sequences are given by the Fuss-Catalan-Raney numbers.
Hankel determinants of linear combinations of moments of orthogonal polynomials, II
本文提出了一个将正交多项式矩的线性组合的 Hankel 行列式表示为正交多项式行列式的公式，给出了四种不同证明及两个应用，涵盖奇异情形计算与经典组合序列的线性递推。
Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the classical (non-consecutive) case, and later by Auli--Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering {\em consecutive patterns of relations}, in analogy to the work of Martinez--Savage in the classical case. Specifically, given two binary re…
In this paper, we extend to a non-consecutive case, the study of the pattern matching condition on the wreath product of the cyclic group and the symmetric group initiated by the authors in a previous paper. The main focus of our paper is (colored) patterns of length 2, although a number of enumerative results for longer patterns are also presented. A new non-trivial bijective interpretation for the Catalan numbers is found, in terms of simultaneously bi-avoiding two patterns in a wreath product.
Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences
本文通过拟算术均值统一研究递增整数序列中缺失整数的和，推广了Barry的算术间隙和到几何与调和情形，并应用于斐波那契和卢卡斯序列，得到调和间隙和指数收敛到ln(α)以及倒数和渐近展开式。
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