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Walks with small steps in the quarter plane
 Contemporary Mathematics
"... Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a mo ..."
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Cited by 54 (8 self)
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Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a halfplane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be Dfinite. The 23rd model, known as Gessel’s walks, has recently been proved by Bostan et al. to have an algebraic (and hence Dfinite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a nonDfinite generating function. Our approach allows us to recover and refine some known results, and also to obtain new
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 45 (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.
Walks in the quarter plane: Kreweras’ algebraic model
, 2004
"... We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: NorthEast, West and South. These walks are known to have remarkable enumerative and probabilistic properties: – they are counted by nice numbers (Kreweras 1965), – t ..."
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Cited by 35 (10 self)
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We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: NorthEast, West and South. These walks are known to have remarkable enumerative and probabilistic properties: – they are counted by nice numbers (Kreweras 1965), – the generating function of these numbers is algebraic (Gessel 1986), – the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function (Flatto and Hahn 1984). These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.
AUTOMATIC CLASSIFICATION OF RESTRICTED LATTICE WALKS
"... Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1. ..."
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Cited by 34 (9 self)
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Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1.
Two Nonholonomic Lattice walks in the Quarter Plane
, 2007
"... We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks w ..."
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Cited by 34 (3 self)
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We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the number of walks of length n.
Polynomial equations with one catalytic variable, algebraic series and map enumeration
 J. Comb. Theory, Series B96
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Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 22 (10 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Lattices of lattice paths
 J. Statist. Plann. Inference
, 2005
"... We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e., lattices of Dyck paths) and give a recursive construction fo ..."
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Cited by 19 (8 self)
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We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e., lattices of Dyck paths) and give a recursive construction for them. The last section is devoted to the presentation of a couple of open problems.