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.
Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking functions. By representing parking functions as labeled digraphs, Hivert, Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on $\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous to the product and coproduct of the Hopf algebra NCSym of symmetric functions in noncommuting variables defined in terms of the power sum basis. In this paper, we view a parking function as a word. We shall construct a Hopf algebra PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogo…
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These obje…
We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We…
We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n+1)^{n-1} in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schroder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to some family of di- or tri-algebras occuring in the works of Loday and Ronco. Moreover, the fundament…
We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $Γ\wr\SG_n$ and the corresponding generalizations of quasi-symmetric functions. Also, we obtain Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type $B$.
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions, and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $Γ\wr\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras …
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and contains the descent algebra as a left ideal.
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structu…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between…
The permutation pattern Hopf algebra is a commutative filtered and connected Hopf algebra. Its product structure stems from counting patterns of a permutation, interpreting the coefficients as permutation quasi-shuffles. The Hopf algebra was shown to be a free commutative algebra and to fit into a general framework of pattern Hopf algebras, via species with restrictions. In this paper we introduce the cancellation-free and grouping-free formula for the antipode of the permutation pattern Hopf algebra. To obtain this formula, we use the popular sign-reversing involution method, by Benedetti and…
We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial. The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainabl…
We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct series of vector spaces whose dimensions are Catalan numbers and Fuss--Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Chered…
There is a very natural and well-behaved Hopf algebra morphism from quasisymmetric functions to peak algebra, which we call it Theta map. This paper focuses on generalizing the peak algebra by constructing generalized Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf subalgebra of any combinatorial Hopf algebra. We also give a combinatorial description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are …
A family of permutations called 2-clumped permutations forms a basis for a sub-Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. The 2-clumped permutations are in bijection with certain decompositions of a square into rectangles, called generic rectangulations. Thus, we can describe the Hopf algebra of 2-clumped permutations using generic rectangulations (we call this isomorphic Hopf algebra the Hopf algebra of generic rectangulations). In this paper, we describe the cover relations in a lattice of generic rectangulations that is a lattice quotient of the right weak order …
We extend the definition of tridendriform bialgebra by introducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q-Gerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of q-tridendriform bialgebras, denoted ST(q) and…
For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We sh…
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generalizations such as the word Hopf algebra and the double word Hopf algebra (and many more).
Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.
In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions QSymm. It has also become clear that it is important to understand the noncommutative versions of such important structures as Symm the Hopf algebra of symmetric functions. Not least because the right noncommmutative versions are often more beautiful than the commutaive ones (not all cluttered up with counting coefficients). NSymm and QSymm are not truly the…
We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Rosas, and Sagan.
By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
For $G$ a finite abelian group, we study the properties of general equivalence relations on $G_n=G^n\rtimes \SG_n$, the wreath product of $G$ with the symmetric group $\SG_n$, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of $\k G_n$ as well as graded connected Hopf subalgebras of $\bigoplus_{n\ge o} \k G_n$. In particular we construct a $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Ho…
Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it genera…
This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …
