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V.1.1 Some Category Theory

The use of category theory to give a clear definition for algebraic structures is presented to introduce the combinatorial species of structure. Introductory books used for this section on category theory are [Sim11, Mac98]. The definitions for combinatorial species of structure can be found in [Joy81, BLL98, FlSe09]. This section aims to provide motivation and diagrams to emphasise the key points of these definitions and their application.

The main thrust of the work by Joyal [Joy81], is to give a precise definition of a labelled structure, through a definition as a functor between the category of finite sets with bijections and itself.

Definition. A Category C consists of:

1. A class Ob(C) whose elements are called objects

2. A class hom(C) whose elements are called morphisms. Each morphism has a source object a and target object b. The set of morphisms from a to b is denotedhom(a, b)

3. A binary operation ◦, called the composition of morphisms. For any three objects a, b and c, we have:

◦:hom(a, b)×hom(b, c)→hom(a, c) (V.1.1)

We write the compositionf◦g or f g. This has to satisfy the two properties: (a) Associativity: If f :a → b, g :b → c and h :c → d, then h◦(g◦f) =

(h◦g)◦f

(b) Identity: For every object x, there exists a morphism Ix : x → x called

the identity morphism, such that for eachf :a→b, f◦Ia=Ib◦f =f Categories also contain the natural idea of a functor: a structure preserving map between categories.

Definition. A (covariant) functor from a category C to a category D written

1. For each object x in C an object F(x) in D

2. For each morphism f :x→y in C a morphismF(f) :F(x)→F(y) in D such that the following two properties hold:

1. For every object x in C, F(Idx) =IdF(x)

2. For all morphismsf :x→y and g:y→z, F(g◦f) =F(g)◦F(f)

V.1.2 The Definitions

Progress on enumerating particular structures on finite sets has been approached by many methods. Some of these methods are:

1. through relations between already enumerated structures 2. through implicit functional equations

3. through recursive relationships - building a structure from other structures on smaller sets

4. through the use of the inclusion-exclusion principle and M¨obius formulæ

The idea to write an exponential generating function, with coefficients being the

number of such structures on a set of sizen, was developed to allow for the techniques

of asymptotic analysis to be used to understand asymptotic enumeration. This is given a thorough exposition in the book by Flajolet and Sedgewick [FlSe09].

The key power of combinatorial species of structure is to relate combinato- rial operations to operations on (exponential) power series. Furthermore, generating functions are a natural concept in probability, where the coefficients represent mo- ments of a random variable. The links between probability and combinatorics are strengthened by this relationship.

A combinatorial species of structure is a functor from the category of finite sets with bijections to itself, as described below:

Definition. A species of structures is a ruleF, which

i) Produces for each finite set U a finite set F[U]

ii) Produces for each bijection σ:U →V a bijection F[σ] :F[U]→F[V]

The functionsF[σ] are required to satisfy the following functorial properties: i) For all bijectionsσ:U →V and τ :V →W, We have: F[τ◦σ] =F[τ]◦F[σ]

ii) For the identity mapIdU :U →U, we haveF[IdU] =IdF[U]

An elements∈F[U]is called an F-structure on U.

Example(Examples of Species of Structure). Five main non-graphical examples of

species of structure used in this thesis are:

1. The SET species E, where for every finite set U we have the set of species of structures is {U}.

2. The PER species S, where for each finite set we have the structures S|U|- the

set of all permutations on U.

3. Species of structure X. For any finite set U, we get {U} if |U| = 1 and ∅

otherwise.

4. The SpeciesE_N - the indicator species of sets of sizeN. We haveE_N[U] ={U}

if|U|=N and ∅ otherwise.

5. The power set or subset species P, gives for U all subsets of U.

The most useful generating function for statistical mechanics is the exponen- tial generating function:

Definition. The exponential generating series of a species of structure F is the

formal power series:

F(x) = ∞ X n=0 fn xn n! (V.1.2)

where fn=|F[n]|, the number of (labelled) F-structures on a set ofn points.

Example (Exponential Generating Series for our Examples). For our examples

above, the exponential generating series are:

1. For SET E, we have that |E[n]|= 1 ∀n, so E(x) =

∞

P

n=0

xn

n! = exp(x)

2. For PER S, we have |S[n]|=n!∀n, so S(x) =

∞

P

n=0

xn= ₁₋1_x

3. For X, we have |X[1]|= 1 and|X[n]|= 0 for n6= 1, so X(x) =x 4. For E_N, we have |E_N[N]|= 1 and |E_N[n]|= 0 for n6=N, so EN(x) = x

N

N!

5. For P, we have |P[n]|= 2n and soP(x) =

∞ P n=0 2n n!x n_{= exp(2x)}

There are two other important related generating functions for species of structure.

If we have unlabelled species of structure, then we don’t have to divide

by the n! for permuting labels and we use the ordinary generating function. The

isomorphism type of a species of structure is an equivalence relation onF[n] with

s∼t if∃π: [n]→[n] such thatF[π](s) =t (V.1.3)

Definition(The Isomorphism Type Generating Series). The isomorphism type gen-

erating series of a species of structureF is the formal power series:

˜ F(x) = ∞ X n=1 ˜ fnxn (V.1.4)

where f˜n is the number of isomorphism types (equivalence classes) of F-structures

on[n].

A further generating series which contains a lot more information about our

species of structure is the cycle index series ZF. For a general permutation σ, its

cycle type is the sequence (σ1, σ2,· · ·), whereσk is the number of cycles of lengthk

in the decomposition ofσ into disjoint cycles. We define the quantities:

Fix σ:={u∈U|σ(u) =u} and fixσ=|Fixσ| (V.1.5)

Definition(Cycle Index Series). The cycle index series of a species of structureF

is the formal power series in infinitely many variables:

ZF(x1, x2,· · ·) = X n≥0 1 n! X σ∈Sn fix F[σ]xσ1 1 x σ2 2 · · · ! (V.1.6)

This is related to Polya’s Hauptsatz, which is applied to Husimi trees and enumerating connected graphs in terms of two-connected graphs in the work of Ford and Uhlenbeck [FoUh56a, FNU56, FoUh56b, FoUh57].

Definition. Let F and Gbe two species of structure. An equipotence α:F →G is

a family of bijectionsαU :F[U]→G[U], for each finite set U.

If there exists an equipotence between two species F and G, then the two species are called equipotent, which we write asF ≡G.

Remark 28. We note thatF(x) =G(x) is equivalent to F ≡G

Definition. An isomorphism of F toGis a family of bijectionsαU :F[U]→G[U],

which obeys the naturality condition:

For any F-structure s∈F[U] andσ :U →V bijection, one must have σ◦αU(s) =αV(σ◦s) (V.1.7)

That is the following diagram commutes: F[U] F[V] G[U] G[V] αU F[σ] αV G[σ]

The two species are then said to be isomorphic and one writesF 'G

Isomorphic species have the same cycle index series and isomorphism type generating series. Equipotent species don’t necessarily share the same series.