Gilbert Labelle

Hypergeometrics and the cost structure of quadtrees

Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral representations akin to Mellin transforms leads to explicit values for various structure constants related to path length, retrieval costs, and storage occupation.

On combinatorial differential equations

Abstract We analyse the solution set of first-order initial value differential problems of the form dy dx = ƒ(x, y), y(0) = 0 in the context of combinatorial species in the sense of A. Joyal ( Adv. in Math. 42 (1981), 1–82). It turns out that the situation is much richer than in the case of formal power series: many non-isomorphic combinatorial solutions are possible for a given problem, although they all have the same underlying generating series. We give many examples of this phenomenon and also elaborate a combinatorial Newton-Raphson iterative scheme for the construction of the solutions. The multidimensional case is treated explicitly.

On asymmetric structures

A given structure is said to be asymmetric if its automorphism group reduces to the identity. The problem of enumerating asymmetric structures (and, more generally, to count structures according to stabilizers) is usually solved by making use of Mobius inversion techniques and symmetric functions in the context of group actions. This method of solution was introduced by Rota (1964, 1969) who defined special classes of polynomials which may be called asymmetry indicator polynomials. Subsequent developments following similar ideas can be found in Stockmeyer (1971), White (1975), Rota, Smith and Sagan (1977, 1980), Kerber (1986). We present here another approach to this problem within the theory of species of structures in the sense of Joyal (1981, 1985, 1986). Every species of structures F contains a sub-species F, called the flat part of F, consisting of all asymmetric F-structures. We introduce an asymmetry indicator series ΓF(x1, x2, x3,…) by means of which we study the correspondence F↦F in connection with the various operations existing in the theory of species of structures. The main result is that the ΓF behaves with respect to the combinatorial operations of sum, product, substitution and differentiation as does the classical cycle indicator series ZF. As a consequence, the asymmetry indicator series can be applied to the systematic classification and enumeration of asymmetric F-structures when the species F is defined (explicitly or recursively) by combinatorial equations. We illustrate the method on particular species (including enriched trees and rooted trees) and a table of ΓF is given for the atomic species concentrated on small cardinalities. Examples show that ΓF contains information independent of that in ZF.

The discrete Green Theorem and some applications in discrete geometry

The discrete version of Greens Theorem and bivariate difference calculus provide a general and unifying framework for the description and generation of incremental algorithms. It may be used to compute various statistics about regions bounded by a finite and closed polygonal path. More specifically, we illustrate its use for designing algorithms computing many statistics about polyominoes, regions whose boundary is encoded by four letter words: area, coordinates of the center of gravity, moment of inertia, set characteristic function, the intersection with a given set of pixels, hook-lengths, higher order moments and also q-statistics for projections.

Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles

Using general methods from the theory of combinatorial species, in the sense of A. Joyal (Adv. in Math. 42 (1981), 1–82), symmetric powers of suitably chosen differential operators are interpreted combinatorially in terms of “eclosions” (bloomings) of certain kinds of points, called “bourgeons” (buds), into certain kinds of structures, called “gerbes” (bundles). This gives rise to a combinatorial setting and simple proof of a general multidimensional power series reversion formula of the Lie-Grobner type (W. Grobner, “Die Lie-Reihen und ihre Anwendungen,” D. Verlag d. Wiss., Berlin, 1960, 1967; “Monatchefte fur Mathematik,” LXVI Bond, 1962). Some related functional equations are also treated and an adaptation of the results to the reversion of cycle index (indicatrix) series, in the sense of Polya-Joyal (Joyal, loc. cit.), is given.

A note on a result of daurat and nivat

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of self-avoiding closed paths, generalizing in this way a recent result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points.

PROPERTIES OF THE CONTOUR PATH OF DISCRETE SETS

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.

Une approche combinatoire pour l'itération de Newton - Raphson

Starting with an approximation @a having a contact of order n with the species A of R-enriched rooted trees (in the sense of Joyal (Advances in Math.42 (1981), 1-82) and Labelle (Advances in Math.42 (1981), 217-247)), a new approximation @a^+, having a contact of order 2n + 2 with A, is deduced by a purely combinatorial argumentation. This provides a combinatorial setting for the classical Newton-Raphson iterative scheme. A generalization involving contacts of higher orders is also developed.

Sur l'Inversion et l'Iteration Continue des Séries Formelles

This paper deals with the composition of normalised formal power series, in one variable, over an arbitrary field K of characteristic zero. A suitable group structure B ⊙ on the set B of polynomial sequences of binomial type is introduced. This group is used first to obtain many formal variants of the classical Lagrange inversion formula (without using any complex integration). Secondly, via one-parameter subgroups of B ⊙ , iteration (i.e., successive composition) of normalised formal power series is studied in detail for arbitrary orders s ∈ K (“continuous” iteration). The case s = −1 coincides with power series inversion. Many new formulas are derived in the course of the text. The end of the work contains suggestions for future research.

Labelled and unlabelled enumeration of k -gonal 2-trees

In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or k-sided polygons (k-gons), where k ≥ 3 is given. This generalization, to k-gonal 2-trees, is natural and is closely related, in the planar case, to some specializations of the cell-growth problem. Our goal is the labelled and unlabelled enumeration of k-gonal 2-trees according to the number n of k-gons. We give explicit formulas in the labelled case, and, in the unlabelled case, recursive and asymptotic formulas.