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50
Basic Analytic Combinatorics of Directed Lattice Paths
 Theoretical Computer Science
, 2001
"... This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then ess ..."
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Cited by 58 (10 self)
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This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.
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 (5 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 confined in a quadrant are not always Dfinite
"... We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z , and always stay in the quadrant x 0; y 0. We first give a criterion which guarantees that the length generating function of these walks is Dfinite, that is, satisfies a li ..."
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Cited by 32 (5 self)
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We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z , and always stay in the quadrant x 0; y 0. We first give a criterion which guarantees that the length generating function of these walks is Dfinite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion
Permutations avoiding an increasing number of lengthincreasing forbidden subsequences
 Discrete Math. Theor. Comput. Sci
, 2000
"... A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Le ..."
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Cited by 23 (1 self)
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A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Let ¢ £ be the set of subsequences of the “¥§¦©¨�������¦©¨����� � form ¥ ”, being any permutation ��������������¨� � on. ¨��� � For the only subsequence in ¢�� ���� � is and ���� � the –avoiding permutations are enumerated by the Catalan numbers; ¨��� � for the subsequences in ¢� � are, ������ � and the (������������������ � –avoiding permutations are enumerated by the Schröder numbers; for each other value ¨ of greater � than the subsequences in ¢ £ ¨� � are and their length ¦©¨����� � is; the permutations avoiding ¨�� these subsequences are enumerated by a number ������ � �� � � sequence such �������������� � that �� � , being � the –th Catalan number. For ¨ each we determine the generating function of permutations avoiding the subsequences in ¢� £ , according to the length, to the number of left minima and of noninversions.
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 22 (7 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.
The insertion encoding of permutations
, 2005
"... We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Appl ..."
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Cited by 18 (3 self)
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We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.
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 12 (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.
The siteperimeter of bargraphs
 Adv. in Appl. Math
"... The siteperimeter enumeration of polyominoes that are both column and rowconvex is a well understood problem that always yields algebraic generating functions. Counting more general families of polyominoes is a far more difficult problem. Here we enumerate (by their siteperimeter) the simplest fa ..."
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Cited by 10 (3 self)
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The siteperimeter enumeration of polyominoes that are both column and rowconvex is a well understood problem that always yields algebraic generating functions. Counting more general families of polyominoes is a far more difficult problem. Here we enumerate (by their siteperimeter) the simplest family of polyominoes that are not fully convex — bargraphs. The generating function we obtain is of a type that, to our knowledge, has never been encountered so far in the combinatorics literature: a qseries into which an algebraic series has been substituted. 1
Walks on the Slit Plane
 Probab. Theory Related Fields
"... In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the halfline H = f(k; 0); k 0g. We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three va ..."
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Cited by 7 (1 self)
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In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the halfline H = f(k; 0); k 0g. We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable generating function is algebraic of degree 8. Moreover, for any point (i; j), the length generating function for walks of this type ending at (i; j) is also algebraic, of degree 2 or 4, and involves the famous Catalan numbers. Our method is based on the solution of a functional equation, established via a simple combinatorial argument. It actually works for more general models, in which walks take their steps in a finite subset of Z 2 satisfying two simple conditions. The corresponding generating functions are always algebraic. In the second part of the paper, we derive from our enumerative results a number of probabilistic corollaries. For instance, we can compute exactly the proba...
Generating functions for the area below some lattice paths
 In Discrete random walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., AC
, 2003
"... We study some lattice paths related to the concept of generating trees. When the matrix associated to this kind of trees is a Riordan array D ¡ d ¡ t¢¤ £ h ¡ t¢¥ ¢ , we are able to find the generating function for the total area below these paths expressed in terms of the functions d ¡ t ¢ and h ¡ ..."
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Cited by 5 (1 self)
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We study some lattice paths related to the concept of generating trees. When the matrix associated to this kind of trees is a Riordan array D ¡ d ¡ t¢¤ £ h ¡ t¢¥ ¢ , we are able to find the generating function for the total area below these paths expressed in terms of the functions d ¡ t ¢ and h ¡ t¢¤¦