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43
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 47 (7 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 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.
THE COMPLETE GENERATING FUNCTION FOR GESSEL WALKS IS ALGEBRAIC
"... Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating ser ..."
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Cited by 43 (10 self)
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Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating series G(t; x, y) = X g(n; i, j)x i y j t n is an algebraic function. n,i,j≥0 1.
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 32 (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.
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 30 (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.
On the functions counting walks with small steps in the quarter plane
 Publ. Math. Inst. Hautes Études Sci
"... Abstract. Models of spatially homogeneous walks in the quarter plane Z 2 + with steps taken from a subset S of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z) ↦ → Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at (i,j) ..."
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Cited by 16 (6 self)
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Abstract. Models of spatially homogeneous walks in the quarter plane Z 2 + with steps taken from a subset S of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z) ↦ → Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at (i,j) ∈ Z 2 + after n steps is studied. For all nonsingular models of walks, the functions x ↦ → Q(x,0;z) and y ↦ → Q(0,y;z) are continued as multivalued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C 2, the interval]0,1/S [ of variation of z splits into two dense subsets such that the functions x ↦ → Q(x,0;z) and y ↦ → Q(0,y;z) are shown to be holonomic for any z from the one of them and nonholonomic for any z from the other. This entails the nonholonomy of (x,y,z) ↦ → Q(x,y;z), and therefore proves a conjecture of BousquetMélou and Mishna in [5].
ON PARTITIONS AVOIDING 3CROSSINGS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 54 (2006), ARTICLE B54E
, 2006
"... A partition on [n] has a crossing if there exists i1 < i2 < j1 < j2 such that i1 and j1 are in the same block, i2 and j2 are in the same block, but i1 and i2 are not in the same block. Recently, Chen et al. refined this classical notion by introducing kcrossings, for any integer k. In thi ..."
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Cited by 16 (5 self)
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A partition on [n] has a crossing if there exists i1 < i2 < j1 < j2 such that i1 and j1 are in the same block, i2 and j2 are in the same block, but i1 and i2 are not in the same block. Recently, Chen et al. refined this classical notion by introducing kcrossings, for any integer k. In this new terminology, a classical crossing is a 2crossing. The number of partitions of [n] avoiding 2crossings is wellknown to be the nth Catalan number Cn = � � 2n n /(n + 1). This raises the question of counting knoncrossing partitions for k ≥ 3. We prove that the sequence counting 3noncrossing partitions is Precursive, that is, satisfies a linear recurrence relation with polynomial coefficients. We give explicitly such a recursion. However, we conjecture that knoncrossing partitions are not Precursive, for k ≥ 4. We obtain similar results for partitions avoiding enhanced 3crossings.
NONDFINITE EXCURSIONS IN THE QUARTER PLANE
"... Abstract. We prove that the sequence (e S n)n≥0 of excursions in the quarter plane corresponding to a nonsingular step set S ⊆ {0, ±1} 2 with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function ..."
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Cited by 13 (1 self)
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Abstract. We prove that the sequence (e S n)n≥0 of excursions in the quarter plane corresponding to a nonsingular step set S ⊆ {0, ±1} 2 with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not Dfinite. Moreover,