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21
Advanced determinant calculus: a complement
 Linear Algebra Appl
"... Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particu ..."
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Cited by 49 (6 self)
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Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.
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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Cited by 37 (0 self)
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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.
The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis
"... Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1. ..."
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Cited by 23 (7 self)
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Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.
Osculating Random Walks on Cylinders
 in Discrete Random Walks, DRW’03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC
, 2003
"... We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is con ..."
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Cited by 10 (3 self)
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We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also consider families of such paths which together cover every edge of the lattice once and visit every vertex twice. Because these paths may touch but not intersect each other and themselves, we call them osculating walks. The ensemble of such families is also known as the dense O(n = 1) model. We consider in particular such paths in a cylindrical geometry, with the cylindrical axis parallel with one of the lattice directions. We formulate a conjecture for the probability that a face of the lattice is surrounded by m distinct osculating paths. For even system sizes we give a conjecture for the probability that a path winds round the cylinder. For odd system sizes we conjecture the probability that a point is visited by a path spanning the infinite length of the cylinder. Finally we conjecture an expression for the asymptotics of a binomial determinant 1
On FPL configurations with four sets of nested arches,” preprint. condmat/0403268
"... The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a, b, c, d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in te ..."
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Cited by 8 (5 self)
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The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a, b, c, d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of nonintersecting lines, the LindströmGesselViennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a, b, c, d) = 1 or 2. AMS Subject Classification (2000): Primary 05A19; Secondary 52C20, 82B20 03/2004 Given a square grid of side n, Fully Packed Loops (FPL) are sets of paths which visit once and only once each of the n2 sites of the grid and exit through every second of the 4n external edges. FPL of a given size fall into connectivity classes, or link patterns, of configurations with a definite set of connectivities between their external edges. The problem of enumerating FPL of a given link pattern is a challenging problem for the combinatorialist, related to alternating sign matrices and other problems of current interest (see [1,2] for reviews). It is also of relevance in statistical mechanics, as it is related by the RazumovStroganov conjecture [3] to the O(1)loop model of percolation, see [4] for references. This paper, which is a continuation of [5], is devoted to a study of FPL configurations with four sets of nested arches. We shall assume the reader to have some familiarity with the ideas and techniques developed in [5] for the case of three sets of nested arches. In particular, with the notion that the boundary conditions force a certain number of edges to be occupied or empty (“fixed edges”), see also [2] for a precursor of this idea and [6] for a recent application to other types of FPL configurations. Our aim is not only to get formulas as explicit as possible for the numbers of these FPL configurations, but also –and mainly – to see to what extent this problem is equivalent to the counting of tilings of certain domains of the triangular lattice, or in a dual picture, to that of dimer configurations on a certain graph. d
On the number of fully packed loop configurations with a fixed associated matching
 ELECTRONIC J. COMBIN
, 2005
"... We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11(1) (2004), Article #R13]. ..."
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Cited by 7 (0 self)
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We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11(1) (2004), Article #R13].
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
"... We present a number of determinant formulae for the number of tilings of various domains ..."
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Cited by 7 (4 self)
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We present a number of determinant formulae for the number of tilings of various domains
Descending plane partitions and rhombus tilings of a hexagon with triangular hole Preprint, arXiv: math.CO/0310188v1
"... Abstract. It is shown that the descending plane partitions of Andrews can be geometrically realized as cyclically symmetric rhombus tilings of a certain hexagon where an equilateral triangle of side length 2 has been removed from its centre. Thus, the lattice structure for descending plane partition ..."
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Cited by 6 (1 self)
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Abstract. It is shown that the descending plane partitions of Andrews can be geometrically realized as cyclically symmetric rhombus tilings of a certain hexagon where an equilateral triangle of side length 2 has been removed from its centre. Thus, the lattice structure for descending plane partitions, as introduced by Mills, Robbins and Rumsey, allows for an elegant visualization. 1. Introduction. Descending
Elliptic enumeration of nonintersecting lattice paths
 J. Combin. Theory Ser. A
"... Abstract. We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elli ..."
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Cited by 5 (2 self)
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Abstract. We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev’s 10V9 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson’s 8φ7 and Dougall’s 7F6 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the 10V9 summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives. 1. Preliminaries We consider lattice paths in the planar integer lattice Z2 consisting of unit horizontal and vertical steps in the positive direction. Given points u and v, we denote