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290
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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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.
Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole
"... . We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enume ..."
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Cited by 23 (9 self)
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. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...
The Horseshoe Estimator for Sparse Signals
, 2008
"... This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, doubleexponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But th ..."
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Cited by 21 (6 self)
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This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, doubleexponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But the horseshoe enjoys a number of advantages over existing approaches, including its robustness, its adaptivity to different sparsity patterns, and its analytical tractability. We prove two theorems that formally characterize both the horseshoe’s adeptness at large outlying signals, and its superefficient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using a combination of real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full Bayesian modelaveraging approach using a discrete mixture prior to model signals and noise.
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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Cited by 20 (13 self)
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We prove a constant term conjecture of Robbins and Zeilberger (J. Cambin.
Expansion around halfinteger values, binomial sums and inverse binomial sums
 J. Math. Phys
, 2004
"... binomial sums ..."
An Apérylike difference equation for Catalan's constant
 The Electronic Journal of Combinatorics
, 2003
"... Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a secondorder difference equation for these forms and their coefficients. As a consequence we deri ..."
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Cited by 16 (4 self)
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Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a secondorder difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan's constant as well as a new continuedfraction expansion for it. Similar arguments are put forward to deduce a secondorder difference equation and a new continued fraction for ζ(4) = π^4/90.
Elementary Derivations of Summation and Transformation Formulas for QSeries
, 1995
"... this paper were presented, along with related exercises, in the author's minicourse on "qSeries" at the Fields Institute miniprogram on "Special Functions, qSeries and Related Topics," June 12 14, 1995. As is customary, we employ the notations used in BHS for the shifted factorial ..."
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Cited by 15 (0 self)
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this paper were presented, along with related exercises, in the author's minicourse on "qSeries" at the Fields Institute miniprogram on "Special Functions, qSeries and Related Topics," June 12 14, 1995. As is customary, we employ the notations used in BHS for the shifted factorial
Theta hypergeometric series
 Proceedings of the NATO ASI Asymptotic Combinatorics with Applications to Mathematical Physics (St
, 2001
"... Abstract. We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variab ..."
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Cited by 15 (3 self)
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Abstract. We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved. Contents
The Number of Centered Lozenge Tilings of a Symmetric Hexagon
"... Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n \Gamma 1, 2n \Gamma 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregu ..."
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Cited by 14 (7 self)
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Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n \Gamma 1, 2n \Gamma 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides a, a and b. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp's conjecture as a corollary of our results.
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 14 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.