This paper develops a unified enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. 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 1-dimensional 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.

Many families of pattern-avoiding 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 123-avoiding 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.

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 D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion

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 non-inversions.

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: North-East, 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. 1

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.

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.

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¢¤¦

In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line 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...

Whereas walks on N with a finite set of jumps were the subject of numerous studies, walks with an infinite number of jumps remain quite rarely studied. Even for relatively well structured models, the classical approach with context-free grammars fails as we deal with rewriting rules over an infinite alphabet. However, several classes of such walks offer a surprising structure: we make here explicit the associated bivariate functions, and give several theorems on their nature (rational, algebraic) via the kernel method or Riordan arrays theory. We give some examples of recent problems in combinatorics or theoretical computer science which lead to such rules.