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12
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
Applications of graphical condensation for enumerating matchings and tilings
, 2003
"... A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then repar ..."
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Cited by 49 (0 self)
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A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then repartitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions.
Determinant Identities And A Generalization Of The Number Of Totally Symmetric SelfComplementary Plane Partitions
"... We prove a constant term conjecture of Robbins and Zeilberger (J. Cambin. ..."
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We prove a constant term conjecture of Robbins and Zeilberger (J. Cambin.
Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten
, 1997
"... Abstract. A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math. 4 (1995), 87–96) that arose in the study of Thue’s method of approximating algebraic numbers. 1. Introduction. In their study [2] of Thue’s method ..."
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Cited by 8 (5 self)
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Abstract. A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math. 4 (1995), 87–96) that arose in the study of Thue’s method of approximating algebraic numbers. 1. Introduction. In their study [2] of Thue’s method of approximating an algebraic number, Bombieri, Hunt, and van der Poorten conjectured two determinant evaluations, one of which can be restated as follows. Conjecture (Bombieri, Hunt, van der Poorten [2, nexttolast paragraph]). Let b, c be nonnegative integers, c ≤ b, and let ∆(b, c) be the determinant of the
Determinants through the looking glass
 Adv. Appl. Math
"... dedicated to dominique foata on his 65th birthday Using a recurrence derived from Dodgson’s Condensation Method, we provide numerous explicit evaluations of determinants. They were all conjectured, and then rigorously proved, by computerassisted methods, that should be amenable to full automation. ..."
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dedicated to dominique foata on his 65th birthday Using a recurrence derived from Dodgson’s Condensation Method, we provide numerous explicit evaluations of determinants. They were all conjectured, and then rigorously proved, by computerassisted methods, that should be amenable to full automation. We also mention a first step towards that goal, our Maple package, DODGSON, that automates the special case of Hankel and Toeplitz hypergeometric determinants. © 2001 Elsevier Science Key Words: Hankel/Toeplitz hypergeometric determinants; Dodgson’s condensation method. This article is motivated by the computation in [1] that was inspired by the short proof [6] of MacMahon’s determinant evaluation [4], using a determinantal identity of Charles Dodgson [2]. Many special cases of the sampled determinants given belowwere independently discovered by Petkov˘sek [5]. For an excellent and detailed survey of existing methods of proofs of determinant identities, see [3]. For any n by n matrix A, let A r�i � j � denote the r by r minor consisting of r contiguous rows and columns of A, starting with row i and column j.
Generalized Fibonacci polynomials and Fibonomial coefficients
, 2013
"... The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined by {} n k = {n}!/({k}!{n−k}!) where {n}! = {1}{2}... {n}. T ..."
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The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined by {} n k = {n}!/({k}!{n−k}!) where {n}! = {1}{2}... {n}. These quotients are also polynomials in s, t and specializations give the ordinary binomial coefficients, the Fibonomial coefficients, and the qbinomial coefficients. We present some of their fundamental properties, including a more general recursion for {n}, an analogue of the binomial theorem, a new proof of the EulerCassini identity in this setting with applications to estimation of tails of series, and valuations when s and t take on integral values. We also study a corresponding analogue of the Catalan numbers. Conjectures and open problems are scattered throughout the paper.
A determinant of the Chudnovskys generalizing the elliptic Frobenius–Stickelberger– Cauchy determinantal identity, Electron
 J. Combin. 7 (2000), Note
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PASCALLIKE DETERMINANTS ARE RECURSIVE
, 2003
"... Let P = [pi,j]i,j≥0 be an infinite matrix whose entries satisfy pi,j = µpi,j−1 + λpi−1,j + νpi−1,j−1 for i, j ≥ 1, and whose first column resp. row satisfy linear recurrences with constant coefficients of orders ρ resp. σ. Then we show that its principal minors dn satisfy dn = ∑δ j=1 cjωjn dn−j wher ..."
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Let P = [pi,j]i,j≥0 be an infinite matrix whose entries satisfy pi,j = µpi,j−1 + λpi−1,j + νpi−1,j−1 for i, j ≥ 1, and whose first column resp. row satisfy linear recurrences with constant coefficients of orders ρ resp. σ. Then we show that its principal minors dn satisfy dn = ∑δ j=1 cjωjn dn−j where cj are constants, ω = λµ + ν, and δ = ( ρ+σ−2) ρ−1 This implies a recent conjecture of Bacher [2].
Advanced Computer Algebra for Determinants
, 2011
"... We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them has been posed by George Andrews in 1980, the other two are by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra method ..."
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We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them has been posed by George Andrews in 1980, the other two are by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger’s original approach even more powerful and allow for addressing a wider variety of determinants. Finally we present, as a challenge problem, a conjecture about a closed form evaluation of Andrews’s determinant. 1
Lieber Opa Paul, ich bin auch ein experimental Scientist!
, 2002
"... Contrary to popular belief, Math is an experimental science. Hence I am an experimental ..."
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Contrary to popular belief, Math is an experimental science. Hence I am an experimental