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125
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.
Around the Razumov–Stroganov conjecture: Proof of a multiparameter sum rule
 E. J. Combi
"... We prove that the sum of entries of the suitably normalized groundstate vector of the O(1) loop model with periodic boundary conditions on a periodic strip of size 2n is equal to the total number of n×n alternating sign matrices. This is done by identifying the state sum of a multiparameter inhomog ..."
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Cited by 43 (19 self)
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We prove that the sum of entries of the suitably normalized groundstate vector of the O(1) loop model with periodic boundary conditions on a periodic strip of size 2n is equal to the total number of n×n alternating sign matrices. This is done by identifying the state sum of a multiparameter inhomogeneous version of the O(1) model with the partition function of the inhomogeneous sixvertex model on a n × n square grid with domain wall boundary conditions. 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.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Loops, matchings and alternatingsign matrices
 DISCR. MATH
, 2008
"... The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge ..."
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Cited by 32 (6 self)
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The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.
Vicious walkers, friendly walkers and Young tableaux: II With a wall
 J. Phys. A: Math. Gen
"... Research supported by the Australian Research Council. ..."
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Cited by 32 (4 self)
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Research supported by the Australian Research Council.
Symmetry classes of alternating sign matrices
, 2000
"... An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, −1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the nonzero entries alternate in sign. The 8element group of symmetries of the square acts in an obvious way on square matrices. For ..."
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Cited by 30 (0 self)
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An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, −1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the nonzero entries alternate in sign. The 8element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions. 1 1
Quantum Knizhnik–Zamolodchikov equation, generalized Razumov–Stroganov sum rules and extended Joseph polynomials
 J. Phys. A
"... We consider the level 1 solution of quantum Knizhnik–Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley–Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpos ..."
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Cited by 30 (12 self)
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We consider the level 1 solution of quantum Knizhnik–Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley–Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects. 09/2007
The many faces of alternatingsign matrices
, 2008
"... I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket ..."
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Cited by 23 (0 self)
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I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket tilings and full packings of loops. (This article has been published in a conference edition of the journal Discrete Mathematics and Theoretical
Thermodynamic limit of the sixvertex model with domain wall boundary conditions
"... We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions ..."
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Cited by 21 (3 self)
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We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bulk free energy. We find that it is different and bears no simple relation with the free energy for periodic boundary conditions. The six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics (alternating sign matrices). This implies new results for the weighted counting for large size alternating sign matrices. Finally we comment on the interpretation of our results, in particular in connection with domino tilings (dimers on a square lattice). 04/2000