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59
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 33 (4 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.
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 25 (2 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.
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 24 (16 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.
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 22 (14 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.
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 multif ..."
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Cited by 10 (6 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.
Combinatorics of the free Baxter algebra
 Electronic Journal of Combinatorics
, 2005
"... We study the free (associative, noncommutative) Baxter algebra on one generator. The first explicit description of this object is due to EbrahimiFard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to trea ..."
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Cited by 9 (0 self)
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We study the free (associative, noncommutative) Baxter algebra on one generator. The first explicit description of this object is due to EbrahimiFard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasiidempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schröder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday’s
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 9 (6 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.
A new method for computing asymptotics of diagonal coefficients of multivariate generating functions
, 2007
"... Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method. ..."
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Cited by 8 (3 self)
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Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method.
The Euler characteristic of the Whitehead automorphism group of a free product
, 2006
"... A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the ..."
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Cited by 7 (4 self)
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A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms Wh( ∗ n i=1 Gi) when the Gi are of finite homological type; Aut( ∗ n i=1 Gi) and Out( ∗ n i=1 Gi) when the Gi are finite; and the palindromic automorphism groups of finite rank free groups.
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 7 (4 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