Structor Research Articles

Enumerating Structures: Applications of the Hyperoctahedral Group in Combinatorial Species

Combinatorial species provide a framework for counting and classifying for counting and classifying combinatorial structures. A species assigns a set of structures to each finite set, respecting the notion of isomorphism. This approach facilitates the enumeration of various combinatorial objects. The hyperoctahedral group, also known as the signed permutation group, consists of permutations of a set of signed elements. This group plays a crucial role in combinatorial algebra, particularly in the enumeration of certain structures, such as various types of trees and graphs. Generating series, both ordinary and exponential, are powerful tools in combinatorial enumeration. Combining these concepts allows for deeper insights into the relationships between structures and their enumerative properties, paving the way for advanced combinatorial theory and applications in various mathematical fields.

Enumeration of bipartite non-crossing geometric graphs

By using the Riordan array method together with combinatorial species theory, we study enumeration problems for geometric graphs which are connected bipartite non-crossing graphs with n + 1 points in convex position. It is performed from two different points of view, one is by the distance between two vertices 1 and n + 1 , and the other one is by the degree of the vertex n + 1 . As a result, we obtain a production matrix for such geometric graphs, and a formula for the number of connected bipartite graphs, which gives an answer to an open question about the connected bipartite non-crossing graphs.

Counting tanglegrams with species

A tanglegram is a pair of binary trees with the same set of leaves. Unlabeled tanglegrams were counted recently by Billey, Konvalinka, and Matsen, who also proposed the problem of counting several variations of unlabeled tanglegrams. We use the theory of combinatorial species to solve these problems.

A species approach to Rota’s twelvefold way

An introduction to Joyal’s theory of combinatorial species is given and through it an alternative view of Rota’s twelvefold way emerges.

On Integral Structure Types

We introduce integral structure types as a categorical analogue of virtual combinatorial species. Integral structure types then categorify power series with possibly negative coefficients in the same way that combinatorial species categorify power series with non-negative rational coefficients. The notion of an operator on combinatorial species naturally extends to integral structure types, and in light of their ‘negativity’ we define the notion of the commutator of two operators on integral structure types. We then extend integral structure types to the setting of stuff types as introduced by Baez and Dolan, and then conclude by using integral structure types to give a combinatorial description for Chern classes of projective hypersurfaces.

Split Graphs: Combinatorial Species and Asymptotics

A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to Bína and Přibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

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 permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: 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.

A Combinatorial Analysis of Tree-Like Sentences

A sentence over a finite alphabet A, is a finite sequence of non-empty words over A. More generally, we define a graphical sentence over A by attaching a non-empty word over A to each arrow and each loop of a connected directed graph (digraph, for short). Each word is written according to the direction of its corresponding arrow or loop. Graphical sentences can be used to encode sets of sentences in a compact way: the readable sentences of a graphical sentence being the sentences corresponding to directed paths in the digraph. We apply combinatorial equations on enriched trees and rooted trees, in the context of combinatorial species and Polya theories, to analyze parameters in classes of tree-like sentences. These are graphical sentences constructed on tree-like digraphs.

Enumeration of Bipartite Graphs and Bipartite Blocks

We use the theory of combinatorial species to count unlabelled bipartite graphs and bipartite blocks (nonseparable or 2-connected graphs). We start with bicolored graphs, which are bipartite graphs that are properly colored in two colors. The two-element group $\mathfrak{S}_2$ acts on these graphs by switching the colors, and connected bipartite graphs are orbits of connected bicolored graphs under this action. From first principles we compute the $\mathfrak{S}_2$-cycle index for bicolored graphs, an extension of the ordinary cycle index, introduced by Henderson, that incorporates the $\mathfrak{S}_2$-action. From this we can compute the $\mathfrak{S}_2$-cycle index for connected bicolored graphs, and then the ordinary cycle index for connected bipartite graphs. The cycle index for connected bipartite graphs allows us, by standard techniques, to count unlabeled bipartite graphs and bipartite blocks.

Coloring Rings in Species

We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

An antipode formula for the natural Hopf algebra of a set operad

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

Γ-Species and the Enumeration of $k$-Trees

We study the class of graphs known as $k$-trees through the lens of Joyal’s theory of combinatorial species (and a extension known as 'Γ-species' which incorporates data about 'structural' group actions). This culminates in a system of recursive functional equations giving the generating function for unlabeled $k$-trees which allows for fast, efficient computation of their numbers. Enumerations up to $k = 10$ and $n = 30$ (for a k-tree with $n + k − 1$ vertices) are included in tables, and Sage code for the general computation is included in an appendix.

Description and generation of permutations containing cycles

The paper proposes a general approach to generating permutations that contain cycles, based on constructive tools introduced to describe combinatorial sets. Different generation problems for permutations of definite class are formulated and solved. A combinatorial set is introduced to define permutations represented as the multiplication of a definite number of cycles. For this set, combinatorial species and associated generating series are constructed.

Species and functors and types, oh my!

The theory of combinatorial species , although invented as a purely mathematical formalism to unify much of combinatorics, can also serve as a powerful and expressive language for talking about data types. With potential applications to automatic test generation, generic programming, and language design, the theory deserves to be much better known in the functional programming community. This paper aims to teach the basic theory of combinatorial species using motivation and examples from the world of functional programming. It also introduces the species library, available on Hackage, which is used to illustrate the concepts introduced and can serve as a platform for continued study and research.

On the arithmetic product of combinatorial species

We introduce two new binary operations on combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series in the context of species. It allows us to introduce the notion of multiplicative species, a lifting to the combinatorial level of the classical notion of multiplicative arithmetic function. Interesting combinatorial constructions are introduced; cloned assemblies of structures, hyper-cloned trees, enriched rectangles, etc. Recent research of Cameron, Gewurz and Merola, about the product action in the context of oligomorphic groups, motivated the introduction of the modified arithmetic product. By using the modified arithmetic product we obtain new enumerative results. We also generalize and simplify some results of Canfield, and Pittel, related to the enumerations of tuples of partitions with the restrictions met.

The cartesian closed bicategory of generalised species of structures

AbstractThe concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature — including of course Joyal's original notion — together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed.

Species Over a Finite Field

We generalize Joyal's theory of species to the case of functors from the groupoid of finite sets to the category of varieties over Fq. These have cycle index series defined by counting fixed points of twisted Frobenius maps. We give an application to configuration spaces.

Economic analysis of a transesophageal echocardiography-guided approach to cardioversion of patients with atrial fibrillation: The ACUTE economic data at eight weeks

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

A shifted asymmetry index series

Let Z F and Γ F be the cycle and asymmetry index series of a combinatorial species F. When G(0)≠0, we have, in general, Γ F∘ G ≠ Γ F ∘ Γ G , in spite of the fact that we always have Z F∘ G = Z F ∘ Z G . We define and study a new index series, denoted Γ F, ξ , which satisfies Γ F,0 = Γ F , Γ F∘ G = Γ F, ξ ∘ Γ G , where ξ= G(0). We include various explicit examples and some applications to the enumeration of labelled and unlabelled substructures and asymmetric substructures.

Application des fonctions symétriques à la résolution d’équations différentielles autonomes combinatoires

We present a method of solving combinatorial differential equations based on the utilization of symmetric functions in the context of combinatorial species introduced by Joyal (see~Adv. in Math. 42 (1981) 1). Our approach relies on an isomorphism between the ring of symmetric functions and the ring of rational set species, whose elements are formal sums involving variables E i , the species of sets of cardinality i. This isomorphism is used to calculate the kernel of the derivative D operator restricted to the above ring. We also study the equation Y′=(1+Y) 2, Y(0)=0 for which we give, apart from the solution L ∗ (nonempty linear orders), another solution in the half-ring of species of sets. The question of the existence of another solution to this equation has been raised (J. Math. Anal. Appl. 113 (2) (1986) 334). Résumé Nous introduisons une approche fondée sur l'utilisation des fonctions symétriques intervenant dans la résolution d’équations différentielles combinatoires dans le contexte des espèces de structures de Joyal (voir (Adv. in Math. 42 (1981) 1). Notre méthode est basée sur un isomorphisme entre l'anneau des fonctions symétriques et l'anneau des espèces ensemblistes rationnelles. Les éléments de cet anneau sont des sommes formelles à coefficients dans Q de monômes formés des variables E i , E i désignant l'espèce des ensembles de cardinal~ i. Nous utilisons cet isomorphisme pour calculer le noyau de l'opérateur de dérivée D restreint à cet anneau. L’équation différentielle Y′=(1+Y) 2, Y(0)=0 sera aussi objet de notre étude. Nous donnons, en plus de la solution L ∗ des listes non vides, une autre solution dans le demi-anneau des espèces ensemblistes. La question de l'existence d'une autre solution dans les espèces a été posé par Labelle (J. Math. Anal. Appl. 113 (2) (1986) 334).