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44
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
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.
SMALL PERMUTATION CLASSES
, 2007
"... We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountab ..."
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Cited by 18 (2 self)
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We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar. We go on to completely characterize the possible subκ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial wellorder, and atomicity (also known as the joint embedding property).
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.
Finitely labeled generating trees and restricted permutations
 Journal of Symbolic Computation
, 2006
"... Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tre ..."
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Cited by 15 (5 self)
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Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of patterns for which this phenomenon occurs. We also present an algorithm — in fact, a special case of an algorithm of Zeilberger — that is guaranteed to find such a generating tree if it exists. 1.
Partially directed paths in a wedge
 Journal of Combinatorial Theory, Series A
"... The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a mo ..."
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Cited by 12 (1 self)
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The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y = ±pX, and an asymmetric wedge defined by the lines Y = pX and Y = 0, where p> 0 is an integer. We prove that the growth constant for all these models is equal to 1+ √ 2, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when p = 1. From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1. The functional recurrences are solved by a variation of the kernel method, which we call the “iterated kernel method”. This method appears to be similar to the obstinate kernel method used by BousquetMélou (see, for example, references [5, 6]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θfunctions, and have natural boundaries on the unit circle.
Statistics on patternavoiding permutations
, 2004
"... This thesis concerns the enumeration of patternavoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics ‘number of fixed points’ and ‘number of excedances’ is the same in 321avoiding as in 132avoiding permutations. This ge ..."
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Cited by 9 (1 self)
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This thesis concerns the enumeration of patternavoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics ‘number of fixed points’ and ‘number of excedances’ is the same in 321avoiding as in 132avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of nonrational generating functions. Next we present a new statisticpreserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of patternavoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations. Then we introduce a bijection between 321 and 132avoiding permutations that preserves
Generating trees and pattern avoidance in alternating permutations
"... We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A2n(2143) of alternating permutations of length 2n avoiding 2143 and the set of standard Young t ..."
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Cited by 9 (1 self)
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We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A2n(2143) of alternating permutations of length 2n avoiding 2143 and the set of standard Young tableaux of shape 〈n, n, n〉, and between the set A2n+1(2143) of alternating permutations of length 2n+1 avoiding 2143 and the set of shifted standard Young tableaux of shape 〈n+2, n+1, n〉. We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof. 1