Inversion of some series of free quasi-symmetric functions

We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form F_{\omega(I)} where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B. C. V. Ung in [Proc. FPSAC'98, Toronto].Comment: 6 page

A Hopf algebra of parking functions

If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions $(f_n)$. We prove that $(f_n)$ is the Frobenius characteristic of the natural permutation representation of \SG_n on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail.Comment: AmsLatex, 14 page

Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasi-symmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via M\"obius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasi-symmetric functions. Our results reveal a close relationship between the structure of the Malvenuto-Reutenauer Hopf algebra and the weak order on the symmetric groups.Comment: 40 pages, 6 .eps figures. Full version of math.CO/0203101. Error in statement of Lemma 2.17 in published version correcte

Differential Operator Specializations of Noncommutative Symmetric Functions

Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by \kzz and \kttzz the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $\alpha \geq 1$, let \cDazz be the unital algebra generated by the differential operators of \kzz which increase the degree in $z$ by at least $\alpha-1$ and \ataz the group of automorphisms $F_t(z)=z-H_t(z)$ of \kttzz with $o(H_t(z))\geq \alpha$ and $H_{t=0}(z)=0$. First, for any fixed $\alpha \geq 1$ and F_t\in \ataz, we introduce five sequences of differential operators of \kzz and show that their generating functions form a $\mathcal N$CS (noncommutative symmetric) system [Z4] over the differential algebra \cDazz. Consequently, by the universal property of the $\mathcal N$CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms \cS_{F_t}: {\mathcal N}Sym \to \cDazz (F_t\in \ataz), which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms \cS_{F_t} above can also be viewed as specializations of NCSFs by the differential operators of \kzz. Secondly, we show that, in both commutative and noncommutative cases, this family \cS_{F_t} (with all $n\geq 1$ and F_t\in \ataz) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.Comment: Latex, 33 pages. Some mistakes and misprints have been correcte