Related papers: The Hopf monoid and the basic invariant of directe…
In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. They show that it is possible to find a combinatorial…
In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on…
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,…
In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on…
We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the…
We study ordered matroids and generalized permutohedra from a Hopf theoretic point of view. Our main object is a Hopf monoid in the vector species of extended generalized permutahedra equipped with an order of the coordinates; this monoid…
Combinatorial Hopf algebras arise in a variety of applications. Recently, Aguiar and Mahajan showed how many well-studied Hopf algebras are closely related to Hopf monoids in species. In this paper, we study Hopf monoids in graphical…
Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode…
We introduce two coproducts on mixed graphs (that is to say graphs with both oriented and unoriented edges), the first one by separation of the vertices into two parts, and the second one given by contraction and extractions of subgraphs.…
We introduce megagreedoids, which generalize polymatroids, megamatroids, and greedoids. We define a quasisymmetric function invariant for a megagreedoid, and show that it has a positive expansion in the basis of fundamental quasisymmetric…
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant…
This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph and the last is the graph-associahedron associated to it.…
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain…
Using the formalism of species and twisted objects, we introduce two structures of cointeracting bialgebras on hypergraphs, induced by two notions of induced sub-hypergraphs. We study the associated unique morphisms of cointeracting…
The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For…
We define cofinite graphs and cofinite groupoids in a unified way that extends the notion of cofinite groups introduced by Hartley. The common underlying structure of all these objects is that they are directed graphs endowed with a certain…
The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's…
We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics.…