Mireille Bousquet-Mélou

Generating functions for generating trees

Generating trees describe conveniently certain families of combinatorial objects: 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) while providing efficient random generation algorithms. In this paper, we investigate the relationship between structural properties of the rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating functions.

A method for the enumeration of various classes of column-convex polygons

Abstract We present a new method that allows to enumerate many classes of column-convex polygons, according to their perimeter, width and area. The first step of this method leads to a functional equation which defines implicitly the generating function for the class of polygons under consideration. The second step consists in solving this equation. We apply systematically our method to all the usual classes of column-convex polygons: thus, we first refine some already known results for parallelogram polygons, directed and convex polygons, and convex polygons, and then we obtain two new results, namely the generating function for column-convex polygons and directed column-convex polygons.

Linear recurrences with constant coefficients: the multivariate case

Abstract While in the univariate case solutions of linear recurrences with constant coefficients have rational generating functions, we show that the multivariate case is much richer: even though initial conditions have rational generating functions, the corresponding solutions can have generating functions which are algebraic but not rational, D-finite but not algebraic, and even non-D-finite.

Lecture Hall Partitions

AbstractWe prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of “lecture hall partitions of length n ” of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Botts formula for the Poincaré series of the affine Coxeter group

Walks in the quarter plane: Kreweras’ algebraic model

\tilde C_n

Walks confined in a quadrant are not always D-finite

, and one direct proof.

New enumerative results on two-dimensional directed animals

Let F(t, u) = F(u) be a formal power series in t with polynomial coefficients in u. Let F1,...,Fk be k formal power series in t, independent of u. Assume all these series are characterized by a polynomial equation P(F(u), F1,...,Fk,t,u) = 0. We 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 hard-particle model on general planar maps.

Counting Walks in the Quarter Plane

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.