We generalize and prove conjectures of Corteel and Lovejoy, related to overpartitions and divisor functions.

We evaluate Hankel determinants of Meixner polynomials associated to the series exp (\sum a [i] z^i/i), where [1],[2]... are the q-integers (Adv. Appl. M., volume Robbins).

Using a symmetrizing operator, we give a new expression for the Omega operator used by MacMahon in Partition Analysis, and given a new life by Andrews (JCTA 2004).

We give a Newton type rational interpolation formula. It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu. Some classical $q$-series identities and bibasic identities are a consequence of it.

We give a dozen formulas concerning Schubert and Grothendieck polynomials, and their interrelations, half of them being new, and most of them interesting. In particular, we explicit the decomposition of Schubert polynomials as positive sums of Grothendieck polynomials, and show that non-commutative Schubert polynomials are obtained by reading the columns of a two-dimensional Cauchy kernel.

We evaluate different Hankel determinants of Rogers-Szego polynomials, and deduce continued fraction expansions.

We comment and prove the formulas that Sylvester gave about the succesive remainders of two polynomials in one variable, in terms of the roots of the two polynomials.

Three elementary self-contained ways of obtaining Schubert polynomials: interpolation, diagonalization of a Cauchy kernel, or coefficients of Yang-Baxter elements.

Litllewood gave expansions of products of the type $\prod 1/(1-ab)$. We show that the method of Littlewood covers several generalizations recently published.

Characteristic classes for flags of vector bundles, Yang-Baxter coefficients and Grothendieck polynomials can be expressed by a simple statistics on alternating-sign matrices.

We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.

We show how to expand a non-symmetric Cauchy kernel 1 /\prod_{i+j< n} (1-x_i y_j) in the basis of Demazure characters. The construction involves using the left and right structure of crystal graphs on words, and mostly reduces to properties of the jeu de taquin.

We define two-parameter families of noncommutative symmetric functions and quasi-symmetric functions, which appear to be the proper analogues of the Macdonald symmetric functions in these settings.

Notes of a course on ``Symmetric functions and Combinatorial Operators on polynomials'', North-Carolina June 01.

We give the generalization of Newton's interpolation formula to several variables, assuming no previous knowledge of Schubert polynomials, but using only simple vanishing conditions. This is an old work with M.P. Schutzenberger.

The Euclidean division of two formal series in one variable produces a sequence of series that we obtain explicitly, remarking that the case where one of the two initial series is 1 is sufficiently generic. As an application, we define a Wronskian of symmetric functions.

The singular locus of a Schubert variety X_{\mu} in the flag variety for GL_n is the union of Schubert varieties X_{\nu}, where \nu runs over a set Sg(\mu) of permutations in S_n. We describe completely the maximal elements of Sg(\mu) under the Bruhat order, thus determining the irreducible components of the singular locus of X_{\mu}

We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of ${\cal W}_{1+\infty}$, to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.

The different varieties of Jack and Macdonald polynomials can be computed using Yang-Baxter relations (in conjunction with a graph associated to the extended affine symmetric group).

For each integer $k$, we introduce a new family of symmetric polynomials, constructed from sums of tableaux using the charge statistic. We conjecture that the Macdonald polynomials indexed by partitions bounded by $k$ expand positively in terms of these polynomials.

We describe tools related to the symmetric group to compute functions of several variables. Many of them have been implemented as a Maple library (ACE) and can be found on the site http://phalanstere.univ-mlv.fr/~ace

We describe how general Grothendieck Polynomials (representatives of Schubert varieties in the Grothendieck ring of flag manifolds) are related to those for Grasmann manifolds, which themselves are deformations of Schur functions. Compile with : latex ws-p8-50x6-00.tex

We show how to read the classical different matrices representing the symmetric group from graphs easy to generate.

We show that the combinatorial description of cumulants by Lehner, in terms of Motzkin paths, can be extended to the description of powers of continued fractions. (submitted to the Seminaire Lotharingien de Combinatoire)

We give a straightforward description and proof of Sylvester's bijection between strict and odd partitions

Using orthogonal divided differences, we study the ring of polynomial as a free module over the invariants of Weyl groups of type D. We apply this description to the cohomology ring of the corresponding flag variety.

We prove an identity about plane partitions, previously conjectured in the study of shifted Jack polynomials (math.CO/9903020). The proof given is using $\lambda$-ring techniques. It would be interesting to obtain a bijective proof.

We stress the importance of addition in the mathematical work of Gian-Carlo Rota (French version and EuroEnglish version).

We give formulas (involving Schur Q-functions) for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (rectangular) matrices which are symmetric or antisymmetric.

We show that the Kazhdan-Lusztig basis elements C_w of the Hecke algebra of the symmetric group, when w corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form (T_i+f_j(v)), where f_j are rational functions.

Enumeration of square-ice models, or alternating-sign matrices lead to the study of Cauchy type determinants of entries (1/(x-y)(qx-y)) or (1/(x-y)(x-y+ g)), where {x} and{y} are two sets of the same cardinal, and q,g are constants. Up to trivial factors, these determinants are symmetric functions in {x} and {y} that we show how to explicit by factorizing them.

We choose products of Schubert polynomials and Q-functions as a basis of the ring of polynomials as a free module over symmetric polynomials in the squares of the variables. We show in particular how to express vertex operators for P and Q Schur functions in terms of divided differences for the hyperoctahedral group. This gives a description of the cohomology ring of a Lagrangian flag manifold.

We show how to express Macdonald operators and creation operators in terms of divided differences and lambda-rings. We obtain a determinantal expression of Macdonald polynomials.

We indicate how Hecke algebras, Yang-Baxter equation, Hall-Littlewood polynomials, Macdonald polynomials, can be traced back to the parameter $y$ that Hirzebruch introduced in his study of Riemann-Roch theorem.

This survey article reviews the structure of monoid of the set of Young tableaux, its poset struture, and the lifting of symmetric functions to the level of the free algebra. This point of view has been developped together with M.P. Schützenberger.

Let P_i(x) be a matrix, obtained from the standard matrix representing the simple transposition (i,i+1) by adding a parameter x in position (i,i). Then reduced products of such matrices parametrize Schubert varieties. Change of parametrizations are polynomial. Moreover, one recovers many classical combinatorial objects (Rothe diagrams, balanced tableaux, ...) from such matrices.

One usually defines the Bruhat order on the symmetric group by subwords of reduced decompositions or by comparison of tableaux (Ehresmann). M.P. Schutzenberger and I prefered to embedd the symmetric group into a distributive lattice. The most powerful method, however, is to use the Kazdhan-Lusztig basis of the Hecke algebra of the symmetric group: vanishing or not of Kazdhan-Lusztig polynomials ensure that permutations are comparable or not. We explicit these polynomials in the case of vexillary permutations.

Divided differences are a powerful tool on functions of several variables. We show how to recover from them the classical interpolation formulas, as well as the multivariate extension of Newton's interpolation. We give a compact and self-contained exposition of Schubert polynomials, as a basis of the free module of polynomials over the ring of symmetric polynomials. Many exercices (in French) are available on request.

Coding permutations as monomials, one obtains a compact expression of representatives of Young's natural idempotents for the symmetric group, or the Hecke algebra.