Gérard Viennot

Binomial determinants, paths, and hook length formulae

Abstract We give a combinatorial interpretation for any minor (or binomial determinant) of the matrix of binomial coefficients. This interpretation involves configurations of nonintersecting paths, and is related to Young tableaux and hook length formulae.

Algebraic languages and polyominoes enumeration

Abstract In this paper, the use of algebraic languages theory in solving an open problem in combinatorics is shown. By constructing a bijection between convex polyominoes and words of an algebraic language, and by solving the corresponding algebraic system, we prove that the number of convex polyominoes with perimeter 2n + 8 is (2n + 11)4 n −4(2n + 1)( 2n n ) .

Heaps of pieces, I : Basic definitions and combinatorial lemmas

We introduce the combinatorial notion of heaps of pieces, which gives a geometric interpretation of the Cartier-Foatas commutation monoid. This theory unifies and simplifies many other works in Combinatorics : bijective proofs in matrix algebra (MacMahon Master theorem, inversion matrix formula, Jacobi identity, Cayley-Hamilton theorem), combinatorial theory for general (formal) orthogonal polynomials, reciprocal of Rogers-Ramanujan identities, graph theory (matching and chromatic polynomials). Heaps may bring new light on classical subjects as poset theory. They are related to other fields as Theoretical Computer Science (parallelism) and Statistical Physics (directed animals problem, lattice gas model with hard-core interactions). Complete proofs and definitions are given in sections 2, 3,4,5. Other sections give a summary of possible applications of heaps.

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilsons 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehlers formula.

Partitions with prescribed hook differences

We investigate partition identities related to off-diagonal hook differences. Our results generalize previous extensions of the Rogers—Ramanujan identities. The identity of the related polynomials with constructs in statistical mechanics is discussed.

Shuffle of parenthesis systems and Baxter permutations

Abstract A formula for the number alternating Baxter permutations is given. The proof of this formula is given by constructing bijection between permutations, trees, and words. This gives also a combinatorial proof of a formula appearing in the enumerative theory of planar maps.

Combinatorial resolution of systems of differential equations. IV. Separation of variables

In the context of the combinatorial theory of ordinary differential equations recently introduced by the authors, a concrete interpretation is given to the classical method of separation of variables. This approach is then extended to more general equations and applied to systems of differential equations with forcing terms.

Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi

Abstract We give the first combinatorial interpretation of the coefficients of the power series of the elliptic Jacobi functions sn, cn and dn. This is done by introducing a new class of permutations enumerated by the Euler numbers and a new index about permutations having the same distribution as the Eulerian numbers.

Enumerative combinatorics and algebraic languages

We give a survey of recent works relating algebraic languages with the combinatorics of planar pictures (i.e. planar maps, animals, polyominoes, secondary structures,…). Such objects are encoded with words. Applications are in enumeration theory, in connection with statistical Physics, molecular Biology, algorithmic complexity and computer graphics drawing.

Algebraic Languages and Polyominoes Enumeration

The purpose of this paper is to show the use of algebraic languages theory in solving an open problem in combinatorics : give a formula for the number of convex polyominoes.