Basic invariants of the Hopf monoid of hypergraphs and its sub-monoids
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 hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
In combinatorics, Hopf structures give an algebraic framework to deal with operations of merging (product) and splitting (coproduct) combinatorial objects. The notion of Hopf algebra is well known and used in combinatorics for over 30 years, and has proved its great strength in various questions (see for example ). More recently, Aguiar and Mahajan defined a notion of Hopf monoid , akin to the notion of Hopf algebra and built on Joyal’s theory of species . Such as in the case of Hopf algebras ,, a useful application of Hopf monoids is to define and compute polynomial invariants, as was put to light by the recent and extensive paper of Aguiar and Ardila . In particular they give a theorem to generate various polynomial invariants and use it to recover the chromatic polynomial of graphs, the Billera-Jia-Reiner polynomial of matroids and the strict order polynomial of posets. Furthermore they also give a way to compute these polynomial invariants on negative integers hence also recovering the different reciprocity theorems associated to these combinatorial objects.
In this paper, we apply Aguiar and Ardila’s theorem to the Hopf monoid of hypergraphs of . We obtain a combinatorial description for the (basic) invariant in terms of colorings of hypergraphs (Theorem 14). While the method of  seems inappropriate to compute on negative integers, we use another approach (rather technical) to get a reciprocity theorem for hypergraphs (Theorem 19). We then use this results to obtain polynomial invariants on sub-monoids of the Hopf monoid of hypergraphs.
2 Definitions and reminders
2.1 Hopf monoids
A vector species consists of the following data.
For each finite set , a vector space .
For each bijection of finite sets , a linear map . These maps should be such that and .
A sub-species of a vector species is a vector species such that for each finite set , is a sub-space of and for each bijection of finite sets , .
A morphism between vector species is a collection of linear maps satisfying the naturality axiom: for each bijection , .
A connected Hopf monoid in vector species is a vector species with that is equipped with product and co-product linear maps
with and disjoint sets, and subject to the following axioms:
Naturality. For each pair of disjoint sets , , each bijection with domain , we have and .
Unitality. For each set , , , and are given by the canonical isomorphisms .
Associativity. For each triplet of pairwise disjoint sets ,, we have: .
Co-associativity. For each triplet of pairwise disjoint sets ,, we have: .
Compatibility. For each pair of disjoint sets , , each pair of disjoint sets , we have the following commutative diagram, where maps to :
A sub-monoid of a Hopf monoid is a sub-species of stable under the product and co-product maps.
The co-opposite Hopf monoid of is the Hopf monoid with opposite co-product: .
A morphism of Hopf monoids in vector species is a morphism of vector species which preserves the products, co-products (compatibility axiom) and the unity (unitality axiom).
We will use the term Hopf monoid for connected Hopf monoid in vector species. A sub-monoid of a Hopf monoid is a itself a Hopf monoid when equipped with the product and co-product maps of . We consider this to be always the case.
A decomposition of a finite set is a sequence of pairwise disjoint subsets such that . A composition of a finite set is a decomposition of without empty parts. We will note for a decomposition of , if is a composition, the length of a decomposition and the number of elements in the decomposition.
Let a be a Hopf monoid. The antipode of M is the morphism of Hopf monoids defined by:
for any finite set .
A character on a Hopf monoid is a collection of linear maps subject to the following axioms.
Naturality. For each bijection we have .
Multiplicativity. For each disjoint sets , we have .
Let us recall from  the results which we will use in the sequel.
Theorem 5 (Proposition 16.1 and Proposition 16.2 in ).
Let be a Hopf monoid and a character on . For and an integer we define:
Then is a polynomial invariant in verifying:
Let be a Hopf monoid. For a set and we call discrete if and for . Then the maps that sends discrete elements onto 1 and other elements onto 0 give us a character of Hopf monoid. Following the terminology introduced in Section 17 of , we call basic invariant of the polynomial invariant obtained by applying Theorem 5 with this character. We note this polynomial or just when is clear from the context.
Proposition 6 (Proposition 16.3 in ).
Let and be two Hopf monoids, and characters on and and a Hopf monoid morphism such that for every :
Denote by and the polynomial invariants obtained by applying Theorem 5 with and and and . For every , one then has:
In particular, since Hopf monoid morphisms conserve discrete elements, for a Hopf monoid morphism and a set, we have .
2.2 A useful combinatorial identity
We remind here a classical result of combinatorics and a direct corollary which will be useful in the following section. We shall only give a sketch of the proofs.
In all the following, given an integer we note .
Let and be two integers. The number of surjections from to is given by:
This formula can be obtained by the inclusion-exclusion principle. ∎
For and two integers such that , and a polynomial of degree at most , we have:
The statement above is a direct consequence of the fact that for . ∎
3 Basic invariant of Hypergraphs
In all the following, always denotes a finite set.
Our goal is to express the basic invariant of the Hopf monoid of hypergraph defined in Section 20 of . More specifically we intend to obtain a combinatorial interpretation of and .
In this context,, an hypergraph over is a collection of (possibly repeated) subsets of , which we call edges444in some references, the terms hyperedge or multiedge is used., containing exactly once. The elements of are then called vertices of and denotes the free vector space of hypergraphs over . Note that two hypergraphs over different sets can never be equal, e.g is not the same as . This is illustrated in Figure 1
The product and co-product are given by, for :
where is the restriction of to and is the contraction of from . The discrete hypergraphs are then the hypergraphs with edges of cardinality at most 1.
For , and , we have:
In , Ardila and Aguiar propose a method to obtain a combinatorial interpretation of any polynomial invariant given by Theorem 5 on negative integers, assuming that we have an interpretation of it on positive integers. Their method consists in using a cancellation-free grouping-free formula for the antipode and the third point of Theorem 5. This approach does not seem to us to be appropriate in the case of hypergraphs. This comes from the fact that there is no simple combinatorial way to describe the summation set of their formula for the antipode in this case. We use instead a different approach: we express the polynomial dependency of in , which we then use to calculate and interpret the resulting formula.
Let us begin by giving a proposition which is needed to show the polynomial dependency of in . For and a sequence of positive integers , we define as a function over the integers given by, for :
Let be integers and note for . Then is a polynomial of degree whose constant coefficient is null and the -th, (for ) coefficient is given by
where and , and the numbers are the Bernoulli numbers with the convention .
We show this by induction over . For the expression of the coefficients gives us the well-known identity . Hence the result is true for . Suppose now the result is true for and let be integers. Denote by the coefficient of . We then have:
This concludes this proof. ∎
Before stating our results on we need to introduce some definitions. There exists a canonical bijection between decompositions and functions with co-domain of the form . In the sequel, we will want to seamlessly pass from one notion to the other. We hence give a few explanations on this bijection. Given an integer , the canonical bijection between decompositions of of size and functions from to is given by:
If it is clear from the context what are and we will write instead of . If is a partition we will also refer to by so that instead of writing ”i such that ” we can just write . Similarly if is a function we will refer to by so that . Also remark that induces a bijection between compositions of of size and surjections from to .
Let be a hypergraph over and be an integer. A coloring of with is a function from to (or a decomposition of of length from what precede) and in this context the elements of are called colors.
Let be a coloring of . For , we say that is a maximal vertex of (for ) if is of maximal color in and we call maximal color of (for ) the color of a maximal vertex of . We say that a vertex is a maximal vertex (for ) if it is a maximal vertex of an edge.
If is a subset of vertices, the order of appearance of (for ) is the composition where . The map sends any decomposition on the composition obtained by dropping the empty parts.
We represent the coloring of a hypergraph on with 1,2,3,4:
The maximal vertex of is and the maximal vertices of are and . The maximal color of is 3. The order of appearance of is The order of appearance of all edges is .
Let be a hypergraph over . An orientation of is a function from to such that for every edge . A directed cycle in an orientation of is a sequence of distinct edges such that . An orientation is acyclic if it does not have any cycle. We note the set of acyclic orientation of .
An orientation of and a coloring of with are said to be compatible if for every . They are said to be strictly compatible if is the unique maximal vertex of .
Let be a set and a hypergraph over . Then is the number of colorings of with such that every edge has only one maximal vertex. This is also the number of strictly compatible pairs of acyclic orientations and colorings with . Furthermore, defining , for every , we have that
where for every , and .
For a decomposition of of size , note . Let be a decomposition of of size . Let be an edge. We then have the equivalence:
Hence, we have that
The equivalence between the colorings such that every edge has only one maximal vertex and the strictly compatible pairs of acyclic orientations and colorings is given by the bijection , where is the unique vertex in such that .
Informally, this formula can be obtained by the following reasoning. To choose a coloring such that every edge has only one maximal vertex, one can proceed in the following:
choose the maximal vertex of each edge (),
choose in which order those vertices appear (),
choose the color of those vertices (), (and notice that the set of such choices is empty if , which allow us to not add this non polynomial dependency in at the previous choice),
choose the colors of the yet uncolored vertices which are in the same edge than a vertex of minimal color in (); then those in the same edge than a vertex of second minimal color in (), etc.
More formally, we show that there exists a bijection between the set of colorings such that every edge has only one maximal vertex and the set
Let be a coloring of interest and define:
such that ,
where is the increasing bijection from to ,
The function not being in would imply that there exists a vertex such that . This is not possible, hence . We also have that because by definition of , implies and is increasing. It is also clear that . The image of is then which is in since for every we must have by definition. Let us now consider , , and . Let be the increasing bijection from to and define by and (it is sufficient since is a partition of ). Let us show that is a coloring of interest. Let be ,
if then ) by definition and so since is increasing,
if then with and so .
We conclude the proof by remarking that the two defined transformations are inverse functions. ∎
The coloring given in Example 12 is not counted in since has two maximal vertices. However by changing the color of to 2 we do obtain a coloring where every edge has only one maximal vertex.
We are now interested in the value of . Let us first state two lemmas.
Let be integers. Then
We proceed by induction on . For , we have
where the second equality comes from the fact that when is an odd number different from one. Suppose now our proposition is true up to . In the proof of Proposition 2 we showed that where is the coefficient of . This gives
where the fifth equality is our induction hypothesis. ∎
Let and be two disjoint sets and and be two compositions. The product of and is the composition . The shuffle product of and is the set .
Let be another composition of . We say that refines and note if with a composition of .
Let be a set and a composition of . We have the identity:
Let furthermore be a directed acyclic graph on and consider the constrained set . We have the more general identity:
Since we only need to show that to prove the first identity. Since the compositions of of size and the surjections from to are in bijection, we have that:
Note that the last equality is a direct consequence of Corollary 8.
To show the second identity first remark that the case where the sum is null is straightforward: if there exists such that , then and so the sum is null. From now on we only consider non empty summation sets. In this case we have that and we only need to show that where . Let us note from now on.
If is not connected let and where and . Let and and suppose without loss of generality that . To choose in we can first choose its length; then which indices are going to have a part of ; and then which indices among them are also are going to have a part of . This leads to: