Quasisymmetric Functions for Nestohedra (original) (raw)
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arXiv (Cornell University), 2017
A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function F (PB) of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron PB associated to a building set B is described as a morphism from the certain combinatorial Hopf algebra of building sets to quasisymmetric functions. We define the q-analog Fq(PB) and derive its determining recurrence relations. The f-polynomial of the nestohedron PB appears as the principal specialization of the quasisymmetric function Fq(PB).
Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra
arXiv: Combinatorics, 2016
Combinatorial Hopf algebras of painted trees exemplify the connections between operads and bialgebras. These trees were introduced recently as examples of how graded Hopf operads can bequeath structure upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of convex polytopes. The faces posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutahedra, and then certain generalized permutahedra: specifically the graph-associahedra of the star graphs which are known collectively as the \emph{stellohedra}, and the graph-associahedra of the fan-graphs. As an aside we show that the stellohedra also appear as certain lifted generalized permutahedra: graph composihedra for complete graphs. Thus our results show how to represent our new algebras using the graph tubings. We...
A combinatorial non-commutative Hopf algebra of graphs
Discrete Mathematics & Theoretical Computer Science
Combinatorics A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.
The Electronic Journal of Combinatorics, 2012
The combinatorial Hopf algebra on building sets BSetBSetBSet extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized Dehn-Sommerville relations for the Hopf algebra BSetBSetBSet, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the mathbfcmathbfd−\mathbf{c}\mathbf{d}-mathbfcmathbfd−index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.
Between graphical zonotope and graph-associahedron
Cornell University - arXiv, 2022
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. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs. In the last section, we study some properties related to H-polytopes.
Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes
Algebraic & Geometric Topology
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 put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutahedra, and then certain generalized permutahedra: specifically the graph-associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutahedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.
Weighted quasisymmetric enumerator for generalized permutohedra
Journal of Algebraic Combinatorics, 2019
We introduce a weighted quasisymmetric enumerator function associated with generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Besides that, it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.
Commutative combinatorial Hopf algebras
Journal of Algebraic Combinatorics, 2008
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 different structures constructed on the same combinatorial objects.
Combinatorial Hopf Algebras and Towers of Algebras—Dimension, Quantization and Functorality
Algebras and Representation Theory, 2012
Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras n≥0 An can be a pair of graded dual Hopf algebras. Hivert and Nzeutchap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower n≥0 An gives rise to graded dual Hopf algebras, then dim(An) = r n n! where r = dim(A 1). In the case r = 1 we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of these constructions.