A determinant property of Catalan numbers

Mays, Michael; Wojciechowski, Jerzy

doi:10.1016/s0012-365x(99)00140-5

Cited by 26 publications

(20 citation statements)

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“…is the sum of squares, cf. [47], which can also be easily seen by application of recursion (2.3). Furthermore, for the matrix A…”

Section: Hankel Matrices and Chebyshev Polynomialsmentioning

confidence: 89%

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Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Tamm¹

2001

Electron. J. Combin.

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Hankel matrices consisting of Catalan numbers have been analyzed by various authors. Desainte-Catherine and Viennot found their determinant to be $\prod_{1 \leq i \leq j \leq k} {{i+j+2n}\over {i+j}}$ and related them to the Bender - Knuth conjecture. The similar determinant formula $\prod_{1 \leq i \leq j \leq k} {{i+j-1+2n}\over {i+j-1}}$ can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficients ${{2m+1} \choose m}$. Generalizing the Catalan numbers in a different direction, it can be shown that determinants of Hankel matrices consisting of numbers ${{1}\over {3m+1}} {{3m+1} \choose m}$ yield an alternate expression of two Mills – Robbins – Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan – like numbers. The well - known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp – Massey algorithm in Coding Theory, which can be applied in order to calculate the coefficients in the three – term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.

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“…is the sum of squares, cf. [47], which can also be easily seen by application of recursion (2.3). Furthermore, for the matrix A…”

Section: Hankel Matrices and Chebyshev Polynomialsmentioning

confidence: 89%

“…In several recent papers (e.g. [2], [47], [54], [62]) these determinants have been studied under various aspects and formulae were given for special parameters. Desainte-Catherine and Viennot in [24] provided the general solution d…”

Section: Hankel Matrices and Chebyshev Polynomialsmentioning

confidence: 99%

Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Tamm¹

2001

Electron. J. Combin.

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“…Using this representation, one can show that H n (C(x)) = 1, H 1 n (C(x)) = 1. Then using induction and the condensation formula (5), one can show (see, e.g., [17]) that…”

Section: Sulanke and Xin's Quadratic Transformationmentioning

confidence: 99%

“…For instance, the sequence of Catalan number: 1,1,2,5,14,42,132,429,1430 In recent years, a considerable amount of work has been devoted to Hankel determinants of path counting numbers, especially for weighted counting of lattice paths with up step (1, 1), level step ( , 0), ≥ 1, and down step (m − 1, −1), m ≥ 2. Many of such Hankel determinants have attractive compact closed formulas, such as that of Catalan numbers [17], Motzkin numbers [1,8], and Schröder numbers [3]. For instance, Motzkin numbers count lattice paths from (0, 0) to (n, 0) with step set {(1, 1), (1, 0), (1, −1)} that never go below the horizontal axis.…”

Section: Introductionmentioning

confidence: 99%

Hankel determinants and shifted periodic continued fractions

Wang

Xin

Zhai³

2019

Advances in Applied Mathematics

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Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set {(1, 1), (q, 0), ( −1, −1)} for integer parameters m, q, . Again shifted periodic continued fractions appear.Mathematic subject classification: Primary 15A15; Secondary 05A15, 11B83.

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“…This result has been proven many times over. The path counting proof was given at least as early as [43], though it also appears in [9,35]. A proof resting on the Cholesky decomposition of C has also been discovered many times.…”

Section: 3mentioning

confidence: 99%

The polyharmonic Dirichlet problem and path counting

Hangelbroek

Lauve

2014

Journal de Mathématiques Pures et Appliquées

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The purpose of this article is to provide a solution to the m-fold Laplace equation in the half space R d + under certain Dirichlet conditions. The solutions we present are a series of m boundary layer potentials. We give explicit formulas for these layer potentials as linear combinations of powers of the Laplacian applied to the Dirichlet data, with coefficients determined by certain path counting problems.In short, we wish to solve Vg = h where V is an m × m matrix of integral operators on R d−1 . We note that this set up has been considered in planar domains for m = 2 by Chen and Zhou [12, Chapter 8] and Costabel and Dauge [13].Our goal is to solve this system explicitly for modest assumptions on the Dirichlet data h. The solution involves using Fourier analysis to recast this as a multiplier problem, and to identify the matrix symbol of the multiplier equation. Namely, we consider the Fourier transformis a Fourier multiplier. This multiplier can be factored µ(ξ) = D 1 (ξ)MD 2 (ξ) as a product of simple diagonal multipliers D 1 and D 2 (having entries that are, essentially, integral powers of |ξ|), and a matrix of constant entries M.The invertibility of the system hinges on the matrix M, which is shown to have a checkerboard pattern: zeros in the odd entries (i.e., M ij = 0 when i + j is odd) and a two-block structure in the even entries. These blocks are Hankel matrices B and C of binomial coefficients and Catalan numbers, respectively. At this point our second objective emerges: the entries in either block count minimal lattice paths in certain planar graphs. In the Catalan case, these are Dyck paths; for the binomial block we have different but related paths.Developing a connection to these lattice paths allows interesting observations about the fundamental matrix M. In particular, its minors count certain non-intersecting paths in slightly modified graphs-these are easily shown to be positive, and thus the underlying matrix is totally non-negative (B and C are totally positive). Moreover, the path counting perspective easily reveals that the determinant of M is a power of 2 (hence M is invertible, hence g is uniquely determined from h). The problem of determining the inverse of M (for all m) is a challenging combinatorial problem, but essential for obtaining the closed form solution g. A solution to this problem can be phrased in terms of path counting, but finding it by this method proves quite difficult. We determine explicit formulas for the entries of M −1 by other means.1.1. Motivation. The motivation for this problem comes from surface spline approximation and interpolation, introduced in the 1970's by Duchon [17] and Meinguet [36] and further developed by many others. This is an approach to multivariate approximation where elementary functions are constructed by taking linear combinations of scattered translates of the fundamental solution φ for ∆ m in R d , perhaps adding a polynomial term of low degree,This is an appealing methodology because such elementary functions can be constructed and evalu...

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A determinant property of Catalan numbers

Cited by 26 publications

References 2 publications

Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Hankel determinants and shifted periodic continued fractions

The polyharmonic Dirichlet problem and path counting

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