Related papers: Theta maps for combinatorial Hopf algebras
The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two…
A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the…
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…
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric…
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear…
The Malvenuto-Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees…
The well-known descent-to-peak map $\Theta_{\mathrm{QSym}}$ for the Hopf algebra of quasisymmetric functions, $\mathrm{QSym}$, and the peak algebra $\Pi$ were originally defined by Stembridge in 1997. We introduce their noncommutative…
We endow the space of rooted planar trees with an structure of Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labelled trees, $n$--trees, increasing planar trees and sorted trees. These…
This article approaches the counting of subgraphs, in terms of signature-type functionals defined over combinatorial Hopf algebras of graphs. Well-known algebraic identities that arise in the context of counting subgraphs are then captured…
In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the…
The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using…
It is known that there is a Hopf algebra structure on the vector space with basis all heap-ordered trees. We give a new bialgebra structure on the space with basis all permutations and show that there is a direct bialgebra isomorphism…
We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric…
The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this…
We show that with the appropriate choice of coproduct, the type B quasisymmetric functions form a Hopf algebra, and the recently introduced type B peak functions form a Hopf subalgebra.
Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids,…
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…
We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of…
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the…
We study the self-dual Hopf algebra $\h\_{\SP}$ of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from $\h\_{\SP}$ to to the Hopf algebra of free quasi-symmetric functions $\FQSym$ given by linear…