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 half-plane: 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 D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function.Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

Let F (t, u) ≡ F (u) be a formal power series in t with polynomial coefficients in u. Let F 1 , . . . , F k be k formal power series in t, independent of u. Assume all these series are characterized by a polynomial equationWe prove that, under a mild hypothesis on the form of this equation, these k + 1 series are algebraic, and we give a strategy to compute a polynomial equation for each of them. This strategy generalizes the so-called kernel method and quadratic method, which apply, respectively, to equations that are linear and quadratic in F (u). Applications include the solution of numerous map enumeration problems, among which the hardparticle model on general planar maps.

The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n ≥ 1, and let σ 0 be a permutation of n having d i cycles of length i, for i ≥ 1. Let m ≥ 2. We prove that the number of m-tuples σ 1 σ m of permutations of n such that• the group generated by σ 1 σ m acts transitively on 1 2 n , • m i=0 c σ i = n m − 1 + 2, where c σ i denotes the number of cycles of σ i , isA one-to-one correspondence relates these m-tuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m = 2, we recover a formula of Tutte for the number of Eulerian maps. The proof relies on the idea that maps are conjugacy classes of trees and extends the method previously applied to Eulerian maps by the second author. Our result might remind the reader of an old theorem of Hurwitz, giving the number of mtuples of transpositions satisfying the above conditions. Indeed, we show that our result implies Hurwitz' theorem. We also briefly discuss its implications for the enumeration of nonequivalent coverings of the sphere.

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and published in its vol. 246(1-3), March 2002, pp. 29-5

No abstract

A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Préa in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them. However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well known to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even D-finite. The fourth class-general prudent walks-is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-D-finite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed. Fig. 1. A self-avoiding walk on the square lattice, and a (quasi-)random SAW of length 1,000,000, constructed by Kennedy using a pivot algorithm [19].

No abstract