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127
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 58 (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.
Proof of the refined alternating sign matrix conjecture
 New York Journal of Mathematics
, 1996
"... Abstract. Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n):= ..."
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Cited by 54 (0 self)
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Abstract. Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n):=
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 46 (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.
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 45 (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
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 43 (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 30 (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
Exact solution of the sixvertex model with domain wall boundary conditions. Critical line between . . .
, 2008
"... This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the presen ..."
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Cited by 30 (7 self)
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This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.
A large dihedral symmetry of the set of alternating sign matrices; www.combinatorics.org/Volume 7/Abstracts/v7i1r37.html, arXiv:math.CO/0006234. 21 G. Xin, private communication
"... We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of paths connecting 2n vertices and show it to be invariant under t ..."
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Cited by 27 (0 self)
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We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of paths connecting 2n vertices and show it to be invariant under the dihedral group D2n rearranging those vertices, which is much bigger than the group of symmetries of the square. We also generalize conjectures of Propp and Wilson relating some of this data for different values of n. 1
A new way to deal with IzerginKorepin determinant at root of unity
, 2002
"... of unity ..."
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