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Advanced determinant calculus: a complement
 LINEAR ALGEBRA APPL
, 2005
"... 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 probl ..."
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Cited by 89 (8 self)
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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.
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 58 (20 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.
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 45 (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.
Enumeration of symmetry classes of alternating sign matrices and characters of classical groups
"... An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variation ..."
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Cited by 39 (3 self)
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An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg’s determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs). 1
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 29 (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
TemperleyLieb stochastic processes
 2002 J.Phys.A:Math.Gen. 35 L661–L668 [arXiv:mathph/0209017
"... Abstract. We discuss onedimensional stochastic processes defined through the TemperleyLieb algebra related to the Q = 1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmet ..."
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Cited by 22 (10 self)
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Abstract. We discuss onedimensional stochastic processes defined through the TemperleyLieb algebra related to the Q = 1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics. PACS numbers: 02.50.Ey, 11.25.Hf, 05.50.+q, 75.10.HkTemperleyLieb Stochastic Processes 2 1.
AN IZERGIN–KOREPINTYPE IDENTITY FOR THE 8VSOS MODEL, WITH APPLICATIONS TO ALTERNATING SIGN MATRICES
, 2008
"... We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin– Korepin formula for the sixvertex model. As applications, we find dynamical (in the sense of the dynamical Yang–Baxter equati ..."
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Cited by 20 (3 self)
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We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin– Korepin formula for the sixvertex model. As applications, we find dynamical (in the sense of the dynamical Yang–Baxter equation) generalizations of the enumeration and 2enumeration of alternating sign matrices.
model with mixed boundary conditions, qKZ equation and alternating sign matrices
 pp, arXiv:mathph/0610067, doi. MR2284004 P. ZinnJustin, UPMC Univ Paris 6, CNRS UMR 7589, LPTHE, 75252 Paris Cedex, France Email address : pzinn@lpthe.jussieu.fr
"... The integrable loop model with mixed boundary conditions based on the 1boundary extended TemperleyLieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of th ..."
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Cited by 19 (1 self)
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The integrable loop model with mixed boundary conditions based on the 1boundary extended TemperleyLieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L + 3. A refined counting is also considered.
Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
, 2008
"... The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at ∆ = −1 2 of size L are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L ..."
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Cited by 16 (3 self)
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The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at ∆ = −1 2 of size L are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L + 1, and knottheoretic skein relations.
From a Polynomial Riemann Hypothesis to Alternating Sign Matrices
 J. Combin
, 2001
"... This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3term recursions. The discussion further leads to higher order polynomial recursions, including 4term recursions where orthogonality is ..."
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Cited by 14 (4 self)
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This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3term recursions. The discussion further leads to higher order polynomial recursions, including 4term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of pn are real and those of pn+1 interleave those of pn ) may be extended to polynomial sequences satisfying certain 4term recursions. We identify speci c polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3term recursions, the 4term recursions give rise naturally to a linear functional. In the case of 3term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4term recursions, we show that the functional can be positive even when there are nonreal zeros among some of the polynomials. It is interesting, however, that for our 4term recursions positivity is guaranteed when a certain real parameter C satis es C 3, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition C 3 is also necessary. Next we used a classical determinant criterion to nd exactly when the associated linear functional is positive, and we found that the Hankel determinants n formed from the sequence of moments of the functional when C = 3 give rise to the initial values of the integer sequence 1; 3; 26; 646; 45885; ; of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 divers...