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137
ChipFiring and RotorRouting on Directed Graphs
, 801
"... Abstract. We give a rigorous and selfcontained survey of the abelian sandpile model and rotorrouter model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems. 1. ..."
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Cited by 70 (18 self)
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Abstract. We give a rigorous and selfcontained survey of the abelian sandpile model and rotorrouter model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems. 1.
The Enumeration of Simple Permutations
 J. Integer Seq
, 2003
"... A simple permutation is one which maps no proper nonsingleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all ..."
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Cited by 47 (4 self)
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A simple permutation is one which maps no proper nonsingleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coe#cients of this series are not P recursive, an asymptotic expansion for these coe# cients, and a number of congruence results.
THE MARKOV CHAIN MONTE CARLO REVOLUTION
"... Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1. ..."
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Cited by 46 (0 self)
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Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1.
Four Classes of PatternAvoiding Permutations under one Roof: Generating Trees with Two Labels
, 2003
"... Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ..."
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Cited by 44 (6 self)
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Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.
Weyl group multiple Dirichlet series II: The Stable Case
"... To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity tha ..."
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Cited by 36 (21 self)
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To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of nth order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.
Exchangeable pairs and Poisson approximation
 Probab. Surv
, 2005
"... This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While ..."
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Cited by 25 (7 self)
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This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered. 1
Random sorting networks
 Adv. in Math
"... A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of Sn generated by nearestneighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the spacetime process of swaps converges to the product of semicircle law and Lebesgue measure. We conj ..."
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Cited by 21 (7 self)
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A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of Sn generated by nearestneighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the spacetime process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at halftime converges to the projected surface measure of the 2sphere. We prove that, in the limit, the trajectories are Hölder1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux. 1
A decomposition of Schur functions and an analogue of the RobinsonSchenstedKnuth algorithm
 Sém. Lothar. Combin
"... Abstract. We exhibit a weightpreserving bijection between semistandard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads ..."
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Cited by 19 (8 self)
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Abstract. We exhibit a weightpreserving bijection between semistandard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the RobinsonSchenstedKnuth Algorithm for skyline augmented fillings. We also prove that the nonsymmetric Schur functions are equal to the standard bases for Schubert polynomials introduced by Lascoux and Schützenberger. This provides a noninductive construction of the standard bases and a simple formula for the right key of a semistandard Young tableau. 1.
Gibbs Fragmentation Trees
, 2008
"... We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the mul ..."
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Cited by 18 (7 self)
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We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the twoparameter Poisson–Dirichlet models for exchangeable random partitions of N, with an extended parameter range 0 ≤ α ≤ 1, θ ≥−2α and α<0, θ =−mα, m ∈ N.