Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species

Transcription

1 Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Department of Mathematics Ira Gessel, Advisor In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by Ji Li August, 2007

2 The signed version of this signature page is on file at the Graduate School of Arts and Sciences at Brandeis University. This dissertation, directed and approved by Ji Li s committee, has been accepted and approved by the Faculty of Brandeis University in partial fulfillment of the requirements for the degree of: DOCTOR OF PHILOSOPHY Adam Jaffe, Dean of Arts and Sciences Dissertation Committee: Ira Gessel, Dept. of Mathematics, Chair. Susan Parker, Dept. of Mathematics Alex Postnikov, Dept. of Mathematics, MIT

3 c Copyright by Ji Li 2007

4 Dedication To Kailang Li and Youzhi Jiang. Lù Màn Màn Qí Xiū Yuăn Xi, Wú Jiāng Shàng Xià Eŕ Qiú Suŏ. by: Qū Yuán The way is long with obstacles, I m questing for the truth to and fro. iv

5 Acknowledgments I would like to thank my advisor, Professor Ira Gessel, for his constant help and support, without which this work would never have existed. I am grateful to the members of my dissertation defense committee, Professor Susan Parker from Brandeis University and Professor Alex Postnikov from MIT. I also owe thanks to the faculty, to my fellow students, and to the kind and supportive staff of the Brandeis Mathematics Department. v

6 Abstract Counting Prime Graphs and Point-Determining Graphs Using Combinatorial Theory of Species A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts by Ji Li In this thesis, we enumerate different types of graphs in light of the combinatorial theory of species initiated by Joyal [13]. The prime graphs with respect to the Cartesian multiplication is enumerated using the exponential composition of species, which is constructed based on the arithmetic product of species studied by Maia and Méndez [18]. The point-determining graphs and bi-point-determining graphs are enumerated by finding functional equations relating them to the species of graphs. vi

7 Contents List of Figures ix Chapter 1. Groups Actions and Pólya s Cycle Index Polynomial Symmetric Groups and Group Actions Pólya s Cycle Index Polynomials Exponentiation Group 8 Chapter 2. Combinatorial Theory of Species Definition of Species Species Operations Molecular Species Compositional Inverse of E Multisort Species Arithmetic Product of Species 38 Chapter 3. Cartesian Product of Graphs and Prime Graphs Introduction Cartesian Product of Graphs Labeled Prime Graphs Unlabeled Prime Graphs Exponential Composition with a Molecular Species Exponential Composition 78 vii

8 3.6. Cycle Index of Prime Graphs 82 Chapter 4. Point-Determining Graphs Introduction Point-Determining Graphs and Co-Point-Determining Graphs Connected Point-Determining Graphs and Connected Co-Point- Determining Graphs Bi-Point-Determining Graphs Colored Graphs and Connected 2-colored Graphs Point-Determining 2-Colored Graphs 117 Appendix A. Index of Species 122 Appendix B. General Notations 124 Appendix. Bibliography 127 viii

9 List of Figures 1.1 Group action of A B Group action of A B Group action of B A Group action of B A An element of B A Cycle type of an element of B A Transport of G -structures Species associated to a graph A linear order and a permutation Addition of species Multiplication of species Composition of species The formula (X m /A) (X n /B) = X m+n /(A B) The formula (X m /A) (X n /B) = X mn /(B A) Unlabeled connected graphs Multisort species The 2-sort species of rooted trees A rectangle. 39 ix

10 2.13 Another rectangle A 3-rectangle Arithmetic product of species Unit of the arithmetic product The species E 2 E The formula (X m /A) (X n /B) = X mn /(A B) The E 2 E 2 -structures An element of A B The Cartesian product of graphs A prime decomposition The automorphism group of a prime power The species E 2 E The formula ((X n /B) m )/A = X nm /B A Unlabeled prime graphs Unlabeled non-prime graphs Labeled point-determining graphs and co-point-determining graphs Transform a graph into a point-determining graph Unlabeled point-determining graphs Unlabeled connected point-determining graphs and unlabeled connected co-point-determining graphs A phylogenetic tree A functional equation for the species of phylogenetic trees. 97 x

11 4.7 Unlabeled phylogenetic trees Unlabeled phylogenetic trees corresponding to the species XE An alternating phylogenetic tree An illustration of (g, h) Operations O Q and O P An illustration of (g, h ) Alternating applications of operations O P and O Q Construct a graph from a given triple (π, ϕ, γ) Unlabeled bi-point-determining graphs The 2-sort species of 2-colored graphs Unlabeled 2-colored graphs counted by the number of edges Unlabeled 2-colored graphs Unlabeled connected 2-colored graphs The formula P s (X, Y ) = (1 + X)(1 + Y ) E (P 2 c (X, Y )) The formula P(X, Y ) = (1 + X + Y ) E (P c 2(X, Y )) Unlabeled point-determining 2-colored graphs Unlabeled point-determining 2-colored graphs with 5 vertices. 121 xi

12 CHAPTER 1 Groups Actions and Pólya s Cycle Index Polynomial 1.1. Symmetric Groups and Group Actions The symmetric group of order n, denoted S n, is the group of permutations of [n] = {1, 2,..., n}. Let λ = (λ 1, λ 2,...), where the λ i are arranged in weakly decreasing order, be a partition of n, denoted λ n. That is, λ = i λ i = n. Let σ be a permutation of [n]. We say λ is the cycle type of σ, denoted by λ = c. t.(σ), if the λ i are the lengths of the cycles in the decomposition of σ into disjoint cycles in weakly decreasing order. Sometimes we write λ = (1 c1(λ), 2 c2(λ),...), where c k (λ) is the number of parts of length k in λ for k 1. Example Let σ = ( ) = (3, 5, 6)(1, 4)(2). Then the cycle type of σ is (3, 2, 1), and c 1 (σ) = c 2 (σ) = c 3 (σ) = 1. It is well-known that the number of permutations of [n] of cycle type λ = (1 c 1, 2 c 2,...,k c k ) is in which the denominator d λ := n!, c 1! 1 c 1 c2! 2 c 2 ck! k c k z λ := c 1! 1 c 1 c 2! 2 c2 c k! k c k 1

13 CHAPTER 1. GROUP ACTIONS is the number of permutations in S n that commute with a permutation of cycle type λ. Definition An action of a group A on a set S is a function ρ : A S S, where for x A and s S, ρ(x, s) is written x s. We say this action is natural if both of the following conditions are satisfied: x (y s) = (xy) s, id A s = s, for any x, y A and s S. Let A be a subgroup of S m, and let B be a subgroup of S n. We construct new groups as described below. Definition We define two groups A B and A B both isomorphic to the product of A and B, where A B is a subgroup of S m+n and A B is a subgroup of S mn. The elements of the product groups are of the form (a, b), where a A and b B. The group operation is defined by (a 1, b 1 ) (a 2, b 2 ) = (a 1 a 2, b 1 b 2 ), where a 1 and a 2 are elements of A, and b 1 and b 2 are elements of B. The group A B acts on the set [m + n] by a(i), (1.1.1) (a, b)(i) = b(i m) + m, if i {1, 2,..., m}, if i {m + 1, m + 2,..., m + n}. 2

14 CHAPTER 1. GROUP ACTIONS Figure 1.1 illustrates the action of an element (a, b) of the group A B. a b Figure 1.1. Group action of A B. The group A B acts on the set [m] [n] and may therefore be viewed as a subgroup of S mn. The action of an element of A B on an element of [m] [n] is given by (a, b)(i, j) = (a(i), b(j)), for all i [m] and j [n]. Figure 1.2 illustrates the action of an element (a, b) of the group A B. b a Figure 1.2. Group action of A B. Definition The wreath product of A and B, denoted B A is the group in which the elements are ordered pairs (α, τ), where α is a permutation in A and τ is 3

15 CHAPTER 1. GROUP ACTIONS a function from [m] to B. An element (α, τ) of B A acts on the set [m] [n] by (α, τ)(i, j) = (αi, τ(i)j), for all i [m] and j [n]. Figure 1.3 illustrates the action of an element (α, τ) of the group B A. α τ(1) τ(2) τ(m) Figure 1.3. Group action of B A. The composition of two elements (α, τ) and (β, η) of B A is given by (α, τ)(β, η) = (αβ, (τ β)η), where β A is viewed as a function from [m] to [m], and (τ β)η denotes the point-wise multiplication of τ β and η, both functions from [m] to B. Again B A can be identified with a subgroup of S mn. Note that the order of B A is B A = A B m. Example Let A be the symmetric group of order 2, and let B be the cyclic group of order 3. The element (α, τ) in B A, where α = (1, 2), τ(1) = id and τ(2) = (1, 2, 3), acts on the set [2] [3] by (1, 1) (2, 1) (2, 1) (1, 2) (1, 2) (2, 2) (2, 2) (1, 3) (1, 3) (2, 3) (2, 3) (1, 1) 4

16 CHAPTER 1. GROUP ACTIONS Therefore, the element (α, τ) of B A is a 6-cycle. The following Theorem is a generalization of the Cauchy-Frobenius Theorem, alias Burnside s Lemma. For the proof of a more general result, with applications and further references, see Robinson [25]. Another application is given in [6]. Theorem (Cauchy-Frobenius) Suppose that a finite group M N acts on a set S. The groups M and N, considered as subgroups of M N, also act on S. The group N acts on the set of M-orbits. Then for any g N, the number of M-orbits fixed by g is given by 1 M fix(f, g), f M where fix(f, g) denotes the number of elements in S that are fixed by (f, g) M N Pólya s Cycle Index Polynomials Definition Let λ be a partition of n. The power sum symmetric function in the variables x 1, x 2,... [27, p. 297] indexed by λ, denoted p λ, is defined by p n = p n [x] = i x n i, n 1 p λ = p λ [x] = p λ1 p λ2 = k 1 p c k(λ) k, if λ = (λ 1, λ 2,...) = (1 c 1(λ), 2 c 2(λ),...). The p λ form a basis for the ring of symmetric functions in the variables x 1, x 2,.... We can also define power sum symmetric functions in the variables y 1, y 2, y 3,..., written as p λ [y], in a similar fashion. Definition The operation called plethysm on the ring of symmetric functions (see Stanley [27, p. 447]) is defined to be such that if f is a symmetric function in the variables x 1, x 2,..., then the plethysm f p n is obtained from the expression for 5

17 CHAPTER 1. GROUP ACTIONS f in terms of the x i s by replacing each x i with x n i. More generally, if f and g are arbitrary symmetric functions, where g is expressed as a function of the power sum symmetric functions, i.e., g = g(p 1, p 2,...), then f g is the polynomial obtained from f by replacing each variable p k by g(p k, p 2k,...). It follows immediately that p m p n = p mn for all positive integers m, n. Next, we introduce an operation on power sum symmetric functions that was studied by Harary [10], and by Maia and Méndez [18]. Definition We define an operation on the symmetric functions by letting p ν := p λ p µ, where c k (ν) = gcd(i, j) c i (λ)c j (µ), lcm(i,j)=k in which lcm(i, j) denotes the least common multiple of i and j, and gcd(i, j) denotes the greatest common divisor of i and j, and then extending it to all symmetric functions through bilinearity. Proposition The operation as defined by Definition is commutative and associative. That is, for any partitions λ, µ and ν, (commutativity) p λ p µ = p µ p λ, (associativity) (p λ p µ ) p ν = p λ (p µ p ν ). Proof. The commutativity is clear. For the associativity, we write (p λ p µ ) p ν = p α, 6

18 CHAPTER 1. GROUP ACTIONS p λ (p µ p ν ) = p β. Using the fact that for any positive integers i and j, ij = lcm(i, j) gcd(i, j), we calculate the number of cycles of length l, for any integer l, in the partition α as follows: c l (α) = lcm(i,j,k)=l = lcm(i,j,k)=l = lcm(i,j,k)=l = lcm(i,j,k)=l gcd(i, j) gcd(lcm(i, j), k) c i (λ)c j (µ)c k (ν) ij lcm(i, j) lcm(i, j) k lcm(lcm(i, j), k) c i(λ)c j (µ)c k (ν) ijk lcm(i, j, k) c i(λ)c j (µ)c k (ν) ijk l c i (λ)c j (µ)c k (ν). The calculation for the partition β is similar. The properties of allow us to talk about the operation on a set of partitions. Let λ (i), i = 1, 2,..., l, be partitions of n 1, n 2,...,n l, respectively. We have p λ (1) p λ (2) p λ (l) = p ν, where ν is a partition of N = n 1 n 2 n l. It is easy to verify that c k (ν) = lcm(j 1,j 2,...,j l )=k l r=1 j r k l ( c ) ji λ (i). i=1 7

19 CHAPTER 1. GROUP ACTIONS Definition Let σ S n. The cycle type monomial of σ, denoted Z(σ), is the monomial in the variables p 1, p 2,...,p n defined by Z(σ) = n k=1 p c k(σ) k = p c.t.(σ). In other words, the cycle type monomial of σ is the power sum symmetric function index by the cycle type of σ. Let A be a subgroup of S n. The cycle index polynomial of A, defined by Pólya [22, pp ], is Z(A) = Z(A; p 1, p 2,...,p n ) = 1 A Z(σ). Pólya [21] (translated in [22]) gave the cycle index polynomials of the product and the wreath product of two permutation groups. σ A Theorem (Pólya) Let A and B be permutation groups on [m] and [n], respectively. Let A B be the product of A and B acting on [m + n], and let B A be the wreath product of A and B acting on [mn]. Then the cycle index polynomials of A B and B A are Z(A B) = Z(A)Z(B), Z(B A) = Z(A) Z(B), where Z(A) Z(B) denotes the plethysm of Z(A) with Z(B) Exponentiation Group Let A be a subgroup of S m, and let B be a subgroup of S n. We introduce in the following another representation of the wreath product (Definition 1.1.4). 8

20 CHAPTER 1. GROUP ACTIONS Definition The exponentiation group B A, studied by Palmer [19], is isomorphic to the wreath product B A. That means, the set of elements of B A is the same as the set of elements of B A, and hence the order of B A is the same as the order of B A, i.e., B A = A B m. However the exponentiation group B A acts on [n] m, identified with the set of functions from [m] to [n], instead of [m] [n]. Given any α A and τ B m, the element (α, τ) B A sends each element f [n] m to the element g [n] m defined by ((α, τ)f)(i) = g(i) = τ(i)(f(α 1 i)), for any i [m]. Therefor, B A can be identified with a subgroup of S n m. Figure 1.4 illustrates the action of an element (α, τ) of the group B A. α α τ(1) τ(5) τ(2) α α τ(4) τ(3) α Figure 1.4. Group action of B A. Example Let A, B, α and τ be the same as in Example

21 CHAPTER 1. GROUP ACTIONS Let f : [2] [3] be such that f(1) = 1 and f(2) = 3. We see that the image of f under the action of (α, τ) B A is the function g : [2] [3] such that g(1) = 3 and g(2) = 2, as shown in Figure 1.5. f(1) = 1 f(2) = 3 g(1) = 3 g(2) = 2 f (α, τ) g Figure 1.5. The element (α, τ) of B A sends f to g. The action of (α, τ) on the set of functions from [2] to [3] is illustrated in Figure 1.6. Hence the cycle type of (α, τ) is (6, 3). Figure 1.6. The cycle type of (α, τ) is (6, 3). The cycle index polynomial of the exponentiation group was given by Palmer and Robinson [20]. They defined the following operators I k for positive integers k. Let R = Q [p 1, p 2,...] be the ring of polynomials with the operation as defined by Definition Palmer and Robinson defined for positive integers k the Q-linear operators I k on R as follows: 10

22 CHAPTER 1. GROUP ACTIONS Let λ = (λ 1, λ 2,...) be a partition of n. The action of I k on the monomial p λ is given by (1.3.1) I k (p λ ) = p γ, where γ = (γ 1, γ 2,...) is the partition of n k with c j (γ) = 1 j ( )( j µ l l j i l/gcd(k,l) ic i (λ)) gcd(k,l). Furthermore, {I k } generates a Q-algebra Ω of Q-linear operators on R. For any elements I, J Ω, any r R and a Q, we set (ai)(r) = a(i(r)), (I + J)(r) = I(r) + J(r), (1.3.2) (IJ)(r) = I(r) J(r). Remark As discussed in Palmer and Robinson s proof of Theorem 1.3.6, if I m (p µ ) = p ν, then ν is the cycle type of an element (α, τ) of the exponentiation group B A acting on [n] m, where α is a permutation in A with a single m-cycle, and τ B m is such that c. t. (τ(m)τ(m 1) τ(2)τ(1)) = µ. Example We take the same A, B, α and τ from Examples That is, A = S 2, B = C 3, α = (1, 2) and τ(1) = id, τ(2) = (1, 2, 3). Then the cycle type of α is λ = (2), and the cycle type of τ(2)τ(1) is µ = (3). The cycle type of (α, τ) acting on the set [3] 2 is ν = (6, 3), as shown in Figure 1.6, hence I 2 (p 3 ) = p 3 p 6. 11

23 CHAPTER 1. GROUP ACTIONS Definition Let f 1 and f 2 be elements of the ring R = Q [p 1, p 2,...]. We define the exponential composition of f 1 and f 2, denoted f 1 f 2, to be the image of f 2 under the operator obtained by substituting the operator I r for the variables p r in f 1. Note that the operation is linear in the left parameters, but not on the right parameters. We call this the partial linearity of the operation. Let A be a subgroup of S m, and let B be a subgroup of S n. Palmer and Robinson proved the following formula [20, pp ] for computing the cycle index polynomial of the exponentiation group B A in terms of the cycle index polynomials of A and B. Theorem (Palmer and Robinson) Let A and B be as described in above. Then the cycle index polynomial of B A is the exponential composition of Z(A) with Z(B). That is, (1.3.3) Z(B A ) = Z(A) Z(B). Example Continuing Example 1.3.4, we apply Theorem to compute the cycle index of the group B A, where A = S 2 and B = C 3. Note that the order of B A is B A = A B 2 = 18. We get the cycle index polynomial of B A Z(B A ) = Z(A) Z(B) [ ] 1 = 2 (p2 1 + p 2) [ ] 1 3 (p p 3) 12

24 CHAPTER 1. GROUP ACTIONS = 1 [ ] 1 2 I 1 3 (p p 3) = 1 [ I 1(p 3 1) I 1(p 3 ) [ ] 1 I 1 3 (p p 3) + 1 [ 1 2 I 2 ] [ 1 3 I 1(p 3 1) I 1(p 3 ) ] 3 (p p 3) ] I 2(p 3 1) I 2(p 3 ) = 1 18 I 1(p 3 1 ) I 1(p 3 1 ) I 1(p 3 ) I 1 (p 3 1 ) I 1(p 3 ) I 1 (p 3 ) I 2(p 3 1 ) I 2(p 3 ) = 1 18 p p p p3 1p p 3p 6 = 1 18 (p p p 3p p 3p 6 ). The cycle index polynomial Z(B A ) tells the structure of the group B A. For example, the coefficient 6 for the term p 3 p 6 tells that there are 6 elements in B A with cycle type (6, 3). These elements are precisely those (α, τ), where α is the 2-cycle in group A, and τ : [2] B is such that τ(1)τ(2) is a 3-cycle in group B. Moreover, we observe that the term p 3 3 in Z(BA ) with coefficient 8 is contributed in two ways. One is from those (α, τ) such that α is the identity element of A, and τ(1) and τ(2) are 3-cycles in B, each having two choices; the other is from those (α, τ) such that α is the identity element of A, one of τ(1) and τ(2) is the identity of B, and the other is a 3-cycle in B, which gives rise to 4 possibilities. Remark Let A and B be the same as in Theorem Let (α, τ) be an element of B A, where c. t.(α) = λ = (λ 1, λ 2,...), and τ B m is such that the cycle type of τ restricted on each λ i -cycle of α is µ (i), i.e., if the i-th cycle of α is written as (a 1, a 2,...,a λi ), 13

25 CHAPTER 1. GROUP ACTIONS then µ (i) is the cycle type of the permutation τ(a 1 )τ(a 2 ) τ(a λi ). Then the cycle type of (α, τ), as an element of the group B A acting on the set of functions from [m] to [n], is ν for which p ν = I λ1 (p µ (1)) I λ2 (p µ (2)). In such cases, for simplicity and without ambiguity, we write p ν = I(λ; µ (1), µ (2),...). 14

26 CHAPTER 2 Combinatorial Theory of Species 2.1. Definition of Species The combinatorial theory of species was initiated by Joyal [13, 12]. In short, species are classes of labeled structures. A formal definition (see Bergeron, Labelle, and Leroux [2, pp. 1 11]) is given in the following. Definition Let B be the category of finite sets with bijections. A species (of structures) is a functor from B to itself, i.e., F : B B. Given a species F, we obtain for each finite set U a finite set F[U], which is called the set of F-structures on U, and for each bijection σ : U V a bijection F[σ] : F[U] F[V ], which is called the transport of F-structures along σ. We denote by F[n] = F[{1, 2,..., n}] the set of F-structures on [n]. The symmetric group S n acts on the set F[n] by transport of structures. The S n -orbits under this action are called unlabeled F-structures of order n. Example The species of graphs is denoted by G. Note that by graphs we mean simple graphs, that is, graphs without loops or multiple edges. Thus defined, 15

27 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES we mean by G [U] the set of graphs with vertex set U, and by a G -structure on U a graph with vertex set U. More formally, the species of structures G generates for any finite set U, a set of graphs with vertex set U; for any bijection σ : U V, a bijection G [σ] : G [U] G [V ]. U = {1, 2, 3, 4, 5} σ 2 4 V = {a,b,c,d,e} a c e b d Figure 2.1. A bijection σ from U to V induces a bijection G [σ], the transport of G -structures along σ, sending a graph with vertex set U to a graph with vertex set V. Example We give a list of examples of species. The empty species 0 is defined by setting 0[U] = for all U. The singleton set species 1 is defined by setting {U}, if U =, 1[U] =, otherwise. The species of singletons X is defined by setting {U}, if U = 1, X[U] =, otherwise. The species of sets E is defined by setting E [U] = {U} for each finite set U. 16

28 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES The species of linear orders L. In particular, the species of linear orders on n-element sets is denoted by X n. The species of connected graphs G c. The species of complete graphs K. The species of permutations Φ. Example For any graph G, we define the species associated to G, denoted O G, to be such that a) for each finite set U, O G [U] is the set of graphs isomorphic to G with vertex set U. Note that L(G) = O G [n]. b) for each bijection σ : U V, where U = V, O G [σ] is a bijection from O G [U] to O G [V ], sending a graph H isomorphic to G with vertex set U to a graph O G [σ](h) whose vertex labeling is obtained from the vertex labeling of H by replacing each label u U with σ(u). It is straightforward to see that the bijections O G [σ] further satisfy i. for all bijections σ : U V and τ : V W, O G [τσ] = O G [τ]o G [σ]. ii. if U = V and σ is the identity map on U, then O G [σ] is also the identity map on O G [U]. a b b c c d d e e a d c e d a e b a c b e a b c d G O G [{a, b, c, d, e}] Figure 2.2. The 5 O G -structures on the vertex set {a, b, c, d, e} for a given graph G. 17

29 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Definition Each species F is associated with three generating series. First, the exponential generating series of the species F, given by counts labeled F-structures. F(x) = n 0 F[n] xn n!, Second, the type generating series of the species F, given by F(x) = n 0 f n x n, where f n is the number of unlabeled F-structures of order n, counts unlabeled F- structures. The last, but also the most important, associated series is called the cycle index of the species F: Z F = Z F (p 1, p 2,...) = n 0 ( λ n fix F[λ] p ) λ. z λ In this definition, fix F[λ] denotes the number of F-structures on [n] fixed by F[σ], where σ is a permutation of [n] with cycle type λ, and p λ is the power sum symmetric function indexed by the partitions λ of n. The following theorem (see [2]) illustrates the importance of the cycle index in the theory of species. Theorem (Bergeron, Labelle and Leroux) For any species of structures F, we have F(x) = Z F (x, 0, 0,...), F(x) = Z F (x, x 2, x 3,...). 18

30 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Remark Consider the functorial aspects of the species of sets E and the species of complete graphs K. We see that there is a natural transformation α that produces for every finite set U a bijection between E [U] and K [U], namely, sending the set U to the complete graph with vertex set U. Furthermore, the following diagram commutes for any finite sets U, V and any bijection σ : U V : E [U] α K [U] E [σ] E [V ] α K [σ] K [V ] In this case we call these two species isomorphic to each other, denoted E = K. The general definition of two species being isomorphic to each other is similar. As pointed out on [2, p. 21], the concept of isomorphism is compatible with the transition to series. Therefore, we do not make distinctions between isomorphic species during calculations. Example For Φ and L the species of permutations and the species of linear orders, we notice that their exponential generating series are identical: Φ(x) = L (x) = 1 1 x, while their type generating series and cycle indices differ: Φ(x) = k 1 Z Φ = k x k, L (x) = 1 x, 1, Z L = 1. 1 p k 1 p 1 This is because that although there are same number of permutations and linear orders on a given finite set U, the permutations and linear orders are not transported 19

31 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES in the same manner along bijections. In fact, a linear order admits only a single automorphism, while a permutation admits many automorphisms in general a) b) Figure 2.3. a) A linear order on [5] admits only a single automorphism. b) A permutation on [5] admits six automorphisms. Example Let G be the species of graphs. It is well-known that the exponential generating series of G is given by G (x) = n 0 2 (n 2) xn n!. The cycle index of G was given on [2, p. 76]: (2.1.1) Z G = n 0 ( λ n fix G [λ] p ) λ, z λ where fix G [λ] = i,j 1 gcd(i, j) c i(λ)c j (λ) 1 2 k 1 (k mod 2) ck(λ), in which c i (λ) denotes the number of parts of length i in λ. The above formulas permit us to write out the first several terms of the associated series of G using Maple: G (x) = 1 + x 1! + 2 x2 2! x8 8! + 8 x3 3! + 64 x4 4! x9 9! x5 5! +, x6 6! x7 7! G (x) = 1 + x + 2x 2 + 4x x x x x x 8 +, 20

32 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES ( 4 Z G = 1 + p 1 + (p p 2) + 3 p p 1p ) 3 p 3 ( p p 2 1p 2 + 2p ) 3 p 1p 3 + p 4 ( p p3 1 p 2 + 8p 1 p p2 1 p p 2p 3 + 2p 1 p ) 5 p 5 +, 2.2. Species Operations We describe in this section some frequently used operations on species, namely, the sum, product, and substitution of species. Readers are referred to [2, pp. 1 58] for more detailed definitions of F 1 + F 2, F 1 F 2, F 1 (F 2 ), F 1 F 2, F 1, F 1 for arbitrary species F 1 and F 2. Definition The sum of F 1 and F 2 is denoted by F 1 +F 2. An F 1 +F 2 -structure on a finite set U is either an F 1 -structure on U or an F 2 -structure on U. F1 + F 2 = or F 1 F 2 Figure 2.4. Addition of species F 1 + F 2. The formulae for F 1 + F 2 are given by (F 1 + F 2 )(x) = F 1 (x) + F 2 (x), ( F 1 + F 2 )(x) = F 1 (x) + F 2 (x), Z F1 +F 2 = Z F1 + Z F2. 21

33 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Each species F gives rise to a canonical decomposition F = F 0 + F 1 + F F n +, where {F n } n 0 is the family of species defined by setting, for each n, { F[U], if U = n, F n [U] =, otherwise. If F = F n, then we say the species F is concentrated on the cardinality n. We denote by F + the species of nonempty F-structures: F + = F 1 + F F n +, F[U], if U 1, F + [U] =, if U = 0. Example The species of sets E and the species of linear orders L can be decomposed as E = E 0 + E 1 + E 2 + = 1 + X + E 2 +. L = L 0 + L 1 + L 2 + = 1 + X + X 2 +. Definition The product of F 1 and F 2 is denoted by F 1 F 2. An F 1 F 2 -structure on a finite set U is of the form (π; f 1, f 2 ), where π is an ordered partition of U with two blocks U 1 and U 2, U i possibly empty, and f i is an F i -structure on U i for each i. The formulae for the associated series of F 1 F 2 are given by (F 1 F 2 )(x) = F 1 (x)f 2 (x), ( F 1 F 2 )(x) = F 1 (x) F 2 (x), 22

34 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Z F1 F 2 = Z F1 Z F2. = F 1 F 2 F 1 F 2 Figure 2.5. Multiplication of species F 1 F 2. Definition The composition of F 1 and F 2 is denoted by F 1 F 2, or equivalently, F 1 (F 2 ). An F 1 (F 2 )-structure on a finite set U is a tuple of the form (π, f, γ), where π is a partition of U, f is an F 1 -structure on π, and γ is a set of F 2 -structures on the blocks of π. F 2 F1 F 2 = F 1 F 2 = F 2 F 2 F 2 F 1 F 2 Figure 2.6. Composition of species F 1 F 2. The formulae for the associated series of F 1 (F 2 ) are given by (F 1 (F 2 ))(x) = F 1 (F 2 (x)), ( F 1 (F 2 ))(x) = Z F1 ( F 2 (x), F 2 (x 2 ),...), 23

35 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Z F1 (F 2 ) = Z F1 Z F2, where is the operation of plethysm on symmetric functions defined by Definition Definition A group A is said to act naturally on a species F if, for any finite set U, there is an A-action ρ U : A F[U] F[U] so that for each bijection σ : U V, the following diagram commutes: ρ U A F[U] F[U] id A F[σ] A F[V ] F[σ] ρ V F[V ] Definition The quotient species (see Bergeron, Labelle, and Leroux [2, p. 159]) of F by A, denoted F/A, is defined based on a group A acting naturally on a species F such that a) for each finite set U, the set of F/A-structures on U is the set of A-orbits of F-structures on U, i.e., (F/A)[U] = F[U]/A, b) for each bijection σ : U V, the transport of structures (F/A)[σ] : F[U]/A F[V ]/A is induced from the bijection F[σ] that sends each orbit of the action of A on F[U] to an orbit of the action of A on F[V ]. 24

36 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES The bijections (F/A)[σ] satisfy the functorial properties (F/A)[τσ] = (F/A)[τ](F/A)[σ], (F/A)[id U ] = id (F/A)[U], because of the functoriality of the species F. The notion of quotient species appeared in [8] and [3] as an important tool in combinatorial enumeration. Example Let E + be the species of nonempty sets. Then an (E + E + )-structure on a finite set U is an ordered pair of nonempty subsets (U 1, U 2 ) whose union equals U. The symmetric group S 2 acts naturally on the subscripts of U i for i = 1, 2, with orbits unordered pairs of such (U 1, U 2 ). Thus we get a quotient species (E + E + )/S 2, the species of partitions into two blocks, denoted Ψ (2). Proposition Let F 1 and F 2 be two species. Let A be a group acting naturally on both F 1 and F 2. Then (2.2.1) (F 1 + F 2 )/A = F 1 /A + F 2 /A. Proof. An F 1 + F 2 -structure on a finite set U is either an F 1 -structure on U or an F 2 -structure on U. It follows immediately that an orbit of A acting on the set of F 1 + F 2 -structures on U is either an orbit of A acting on the set F 1 [U] or an orbit of A acting on the set F 2 [U] Molecular Species Definition A species of structures M is said to be molecular [31, 30] if there is only one isomorphism class of M-structures, i.e., if two arbitrary M-structures are isomorphic. 25

37 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES In other words, the species M is molecular if and only if M 0, M is concentrated on n, and any element in the set M[n] can be sent through some transport of structures to any other element in M[n]. We introduce in the following a construction of, for any subgroup A of S n, a molecular species X n /A (see Bergeron, Labelle, and Leroux [2, p. 144]). These species satisfy that X n /A = X n /B if and only if A and B are conjugates in S n. In fact, if M is any molecular species concentrated on the cardinality n, and s is any M-structure, then M is the same as the species X n /A, as defined in below, for some A that is the automorphism group of s. Definition Let n be a non-negative integer, A a subgroup of S n, U and V finite sets of cardinality n, and τ : U V a bijection. Define a species X n /A to be such that a) the set of X n /A-structures on U is the set of (left) cosets with respect to A of bijections [n] U. That is, (X n /A)[U] = {σa σ : [n] U}, where σa = {σ a a A}. b) the transport (X n /A)[τ] : (X n /A)[U] (X n /A)[V ] is defined by setting (X n /A)[τ](σA) = (τσ)a. Example a) There is only one element in the set E n [n], so that the condition any two elements in E n [n] are isomorphic is automatically true. Hence the species 26

38 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES E n is a molecular species, and E n = Xn S n. b) We denote by C n the species of oriented cycles of length n. We see that C n is a molecular species, and C n = Xn C n, where C n is the cyclic group of order n, i.e., C n is the subgroup of S n in which all elements are n-cycles. Remark Let U be a finite set of cardinality n, and let A be a subgroup of S n. Then A acts naturally on the set of linear orders on U. If we call the orbit of a linear order s on U under this A-action the A-orbit of s, then the X n /A-structures on U is just the set of A-orbits of the action of A on the set of linear orders on U, which is also, as pointed out on [2, p. 160], the quotient species of X n by A. The following proposition (see [15, p. 117] Example 7.4) illustrates the close connection between Pólya s cycle index polynomial and the cycle index of a species. Proposition Let A be a subgroup of S n. Then Z(A) = Z X n /A. Proof. Recall that the set (X n /A)[n] is the set of cosets of A in S n. We set (X n /A)[n] = {g 1 A, g 2 A,...,g k A}, where g 1, g 2,...,g k are coset representatives. Then according to the definition of cycle index series, we have Z X n /A = 1 fix Xn [τ] Z(τ), n! A τ S n 27

39 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES where Z(τ) is the cycle type polynomial of τ, and the number of elements in (X n /A)[n] fixed by τ is fix Xn A [τ] = {i [k] : τg ia = g i A} = {i [k] : g 1 i τg i A}. We notice that if for some i, g 1 i τg i A, then every element g in the coset represented by g i A satisfies g 1 τg A. We notice also that each coset g i A contains exactly A elements. Hence we get Therefore, fix Xn A [τ] = 1 {g Sn : g 1 τg A}. A Z X n /A = 1 n! = 1 n! 1 A 1 A τ S n τ, g Sn g 1 τg A {g Sn : g 1 τg A} Z(τ) Z(τ) = 1 A g Sn σ A 1 n! Z(σ) = 1 A Z(σ) = Z(A). σ A Each subgroup A of S n corresponds to a molecular species concentrated on the cardinality n, namely, X n /A. Proposition says that Pólya s cycle index polynomial of A is the same as the cycle index of the species X n /A, hence these two definitions are equivalent in the case of molecular species. Example There are d λ = n!/z λ permutations with cycle type λ in S n. Thus the cycle index polynomial of S n, which is the same as the cycle index of the species 28

40 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES E n, is Z En = Z(S n ) = 1 d λ p λ = n! λ n λ n Example Recall for any graph G, O G is the species associated to G. We oberve that O G is indeed the molecular species X n / aut(g), where n is the number of vertices in G, and aut(g) is the automorphism group of G. It follows from Proposition that the cycle index of O G is the cycle index polynomial of the automorphism group of G. That is, p λ z λ. Z OG = Z(aut(G)). For example, the cycle index of the graph G in Figure 2.2 is Z OG = Z(A), where A is the automorphism group of G, which is the subgroup of S 5 isomorphic to S 4. To be more precise, Z OG = 1 24 (p p3 1 p 2 + 8p 2 1 p 3 + 3p 1 p p 1p 4 ). We see from Definition that molecular species are indecomposable under addition. This observation leads to a molecular decomposition of any species [2, p. 141]: Proposition Every species of structures F is the sum of its molecular subspecies: F = M F M molecular M. 29

41 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Example The molecular decomposition of the species of graphs G is given on [2, p. 421]: G = 1+X +2E 2 +(2E 3 +2XE 2 )+(2E 4 +2E 2 E 2 +2XE 3 +2E X2 E 2 +E X 2 )+ Consider the molecular species X m /A and X n /B. That means A is subgroup of S m and B is a subgroup of S n. Yeh [31, 30] proved the following theorem for molecular species. Theorem (Yeh) Let A be a subgroup of S m, and B a subgroup of S n with n 1. We have X m A Xn B = Xm+n A B, X m A Xn B = Xmn B A, where the product group A B acts on the set [m + n], and the wreath product group B A, as defined by Definition 1.1.4, acts on the set [mn]. Note that Yeh s results agree with Pólya s Theorem for the cycle index polynomials of A B and B A. In fact, as mentioned in Remark 2.3.4, we think of an X m /A-structure on the set U 1 with U 1 = m as an A-orbit of linear orders on U 1, and an X n /B-structure on the set U 2 with U 2 = n as a B-orbit of linear orders on U 2. Then an (X m /A) (X n /B)- structure on U = U 1 U 2, where denotes the disjoint union admits an automorphism group isomorphic to the product group A B, as shown in Figure 2.7. We also observe that an (X m /A) (X n /B)-structure is an X m /A-structure on a set of X n /B-structures, which is the same as a set of B-orbits of linear orders on [n], 30

42 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES A-orbits B-orbits Figure 2.7. (X m /A) (X n /B) = X m+n /(A B). as shown in Figure 2.8, and hence admits an automorphism group isomorphic to the wreath product B A. A-orbits B-orbits B-orbits B-orbits Figure 2.8. (X m /A) (X n /B) = X mn /(B A) Compositional Inverse of E + The species 1 + X is the sum of the singleton set species 1 and the species of singletons X. A 1 + X-structure on [n] is { }, if n = 0, (1 + X)[n] = {1}, if n = 1,, otherwise. 31

43 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES The associated series of 1 + X are (1 + X)(x) = 1 + x, 1 + X(x) = 1 + x, Z (1+X) = 1 + p 1. A virtual species is, roughly speaking, the formal difference of species [2, p. 121]. Since the species 1 + X satisfies (1 + X)(0) = 1, Proposition 18 of [2, p. 129] gives that there exists a unique virtual species Γ, so-called connected (1 + X)-structures as defined on [2, p. 121], with (2.4.1) 1 + X = E (Γ). In fact, Equation (2.4.1) leads to X = E + (Γ), which implies that Γ is the compositional inverse of E +, written as Γ = E 1 + (X). The associated series of Γ are given by [2, p. 131]: Γ(x) = log(1 + X)(x) = log(1 + x), Γ(x) = k 1 Z Γ = k 1 µ(k) k log (1 + X)(x k ) = µ(k) k log(1 + xk ), k 1 µ(k) k log Z (1+X) p k = µ(k) k log(1 + p k), k 1 32

44 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES where µ denotes the Möbius function, defined by 0, if n has one or more repeated prime factors, µ(k) = 1, if n = 1, ( 1) j, if n is a product of j distinct primes. In fact, we can compute the first several terms of the associated series of Γ: Γ(x) = x 1! x2 2! + 2 x3 3! 6 x4 x5 x6 x ! 5! 6! 7!..., (2.4.2) Γ(x) = x x 2, ( 1 Z Γ = p 1 2 p ) ( 1 2 p p3 1 1 ) ( 1 3 p 3 4 p4 1 1 ) ( 1 4 p p5 1 1 ) 5 p 5 +. Read [23, p. 4] derived identity (2.4.2) as the type generating series of the compositional inverse of the species E +. Example Let G c be the species of connected graphs, and E the species of sets. The observation that every graph is a set of connected graphs gives rise to the following species identity: (2.4.3) G = E (G c ), which can be read as a graph is a set of connected graphs, and gives rise to the identity G c = Γ(G + ), which leads to the enumeration of connected graphs: G c (x) = log(g (x)), 33

45 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES G c (x) = k 1 Z G c = k 1 µ(k) k µ(k) k log( G (x k )), log(z G p k p k + 1). Using Maple, we can compute the first several terms of the associated series of G c : (2.4.4) (2.4.5) (2.4.6) G c (x) = x 1! + x2 2! + 4 x3 3! x8 8! + 38 x4 4! x5 5! x9 9! x6 6! +..., x7 7! G c (x) = x + x 2 + 2x 3 + 6x x x x x 8 +, ( 1 Z G c = p p ) ( 1 2 p p ) 3 p3 1 + p 1 p 2 ( p p2 1 p p p 1p ) 2 p 4 ( p 3 1p p 2p p p 1 p p2 1p ) 5 p 5 + p 1 p 4 ( p p4 1 p p3 1 p p2 1 p p2 1 p p 1p 2 p p 1p p p 2p p ) 6 p 6 +. Recall Proposition that states that there is a molecular decomposition of any species up to isomorphism. We can often identify a given species in terms of molecular species using combinatorial operators. Figure 2.9 shows unlabeled connected graphs on no more than 4 vertices. 34

46 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Figure 2.9. Unlabeled connected graphs on n vertices, n 4. In this way, the molecular decomposition of the species of connected graphs G c takes the form ) ) G c = X + E 2 + (XE 2 + E 3 + (E 2 X 2 + XE 3 + X 2 E 2 + E 2 E 2 + E 4 + D 4 +, where D 4 is the molecular species corresponding to the dihedral group D 4 of order 4, that is, D 4 = X 4 /D 4, and Z D4 = Z(D 4 ) = 1 8 (p p 21p p p 4 ) Multisort Species The theory of multisort species [2, p. 100] is analogous to multivariate functions. Definition A k-set U is a k-tuple of sets U = (U 1, U 2,...,U k ). A multifunction f from (U 1, U 2,...,U k ) to (V 1, V 2,...,V k ), denoted f : (U 1, U 2,...,U k ) (V 1, V 2,...,V k ), 35

47 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES is a k-tuple of functions f = (f 1, f 2,...,f k ) such that f i : U i V i for i = 1,..., k. Such f is called bijective if each f i is a bijection. Let B k be the category of k-sets with bijective multifunctions. Definition A species of k sorts, where k 1, is a functor from the category of finite k-sets B k to the category of finite sets B, i.e., F : B k B. A k-sort species F gives rise for each k-set U = (U 1, U 2,...,U k ) a finite set F[U 1, U 2,...,U k ], and for each bijective multifunction σ = (σ 1, σ 2,..., σ k ) : (U 1, U 2,..., U k ) (V 1, V 2,...,V k ) a bijection F[σ] = F[σ 1, σ 2,...,σ k ] : F[U 1, U 2,...,U k ] F[V 1, V 2,...,V k ] that is called the transport of F-structures along σ. We denote by F(X, Y ) a 2-sort species F. We use the notation [m, n], where m and n are positive integers, to denote the 2-set ({1, 2, 3,..., m}, {1, 2, 3,..., n}). The exponential generating series of F(X, Y ) is F(x, y) = m,n 0 F[m, n] x m y n m! n!, where F[m, n] = F[{1, 2,..., m}, {1, 2,..., n}] is the number of labeled F-structures on the 2-set [m, n]. 36

48 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES F Figure Multisort species. The type generating series of F(X, Y ) is F(x, y) = f m,n x m y n, m,n 0 where f m,n is the number of unlabeled F-structures in which the cardinality of the first sort set is m and the cardinality of the second sort set is n. The cycle index of F(X, Y ) is Z F(X,Y ) = ( m,n 0 λ m, µ n fix F[λ, µ] p λ[x] z λ ) p µ [y], z µ where p λ [x] and p λ [y] are power sum symmetric functions in variable sets x 1, x 2,... and y 1, y 2,..., respectively, fix F[λ, µ] denotes the number of F-structures on [m, n] fixed by F[σ, π], where σ is a permutation of [m] with cycle type λ and π is a permutation of [n] with cycle type µ. Example Let A (X, Y ) be the 2-sort species of rooted trees whose internal vertices are of sort X and whose leaves are of sort Y. Figure 2.11 shows the transport 37

49 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES of structures of two A (X, Y )-structures under a bijective multifunction σ = ( 2 j 3 4 a g 5 m 7 f 8 e 9 12 d n 1 l 6 k 10 h 11 b 13 i 14 c ) : 3 a A [σ] f e m d j n g h c l i k b Figure The transport of structures A [σ] sends an element of A [U 1, U 2 ] to an element of A [V 1, V 2 ], where U 1 = {2, 3, 4, 5, 7, 8, 9, 12}, U 2 = {1, 6, 10, 11, 13, 14}, V 1 = {j, a, g, m, f, e, d, n}, V 2 = {l, k, h, b, i, c} Arithmetic Product of Species The arithmetic product was studied by Manuel Maia and Miguel Méndez [18]. They developed a decomposition of a set, called a rectangle, which, as we shall see, is essentially an equivalence class of high dimensional linear orders. Definition Let U be a finite set. Suppose U = ab. A rectangle on U of height a is a pair (π 1, π 2 ) such that π 1 is a partition of U with a blocks, each of size b, and π 2 is a partition of U with b blocks, each of size a, and if B is a block of π 1 and B is a block of π 2 then B B = 1. 38

50 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Figure 2.12 shows two representations of a rectangle on [12] of height 3. They both represent the rectangle (π 1, π 2 ). π 1 = {{1, 3, 4, 9}, {2, 5, 8, 10}, {6, 7, 11, 12}}, π 2 = {{1, 8, 12}, {3, 5, 6}, {2, 4, 11}, {7, 9, 10}} Figure A rectangle on [12] of height 3. Equivalently we can represent a rectangle as a bipartite graph. A rectangle which is the same as in Figure 2.12 is shown by Figure 2.13, where the labels are on the A 1 A B 1 B 2 π 1 = {A 1, A 2, A 3 } A B 3 B 4 π2 = {B 1, B 2, B 3, B 4 } Figure A rectangle on [12] of height 3. edges, and the vertex sets on the left and on the right represent partitions π 1 and π 2. More precisely, each edge x in the bipartite graph corresponds to exactly one pair of vertices (A i, B j ) with A i π 1 and B j π 2, and this can be interpreted as x being the only element in the intersection of A i and B j. 39

51 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES Definition Let U be a finite set. A k-rectangle on U is a k-tuple of partitions (π 1, π 2,...,π k ) such that a) for each i [k], π i has a i blocks, each of size U /a i, where U = k a i. b) for any k-tuple (B 1, B 2,..., B k ), where B i is a block of π i for each i [k], we have B 1 B 2 B k = 1. Figure 2.14 shows a 3-rectangle (π 1, π 2, π 3 ), represented by a 3-partite graph, and labeled are on the triangles. i=1 A 1 A 2 A 3 A π 1 = {A 1, A 2, A 3, A 4 } B π 2 = {B 1, B 2, B 3 } C π 3 = {C 1, C 2 } B C 1 B 3 Figure A 3-rectangle labeled on the triangles. We denote by N the species of rectangles, and by N (k) the species of k-rectangles. Let n = k i=1 a i, and let be the set of bijections of the form δ : [a 1 ] [a 2 ] [a k ] [n]. Note that the cardinality of the set is n!. The group k S ai = {σ = (σ 1, σ 2,...,σ k ) : σ i S ai } i=1 acts on the set by setting (σ δ)(i 1, i 2,...,i k ) = δ(σ 1 (i 1 ), σ 2 (i 2 ),...,σ k (i k )), 40

52 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES for each (i 1, i 2,...,i k ) [a 1 ] [a 2 ] [a k ]. We observe that this group action result in a set of k i=1 S a i -orbits, and that each orbit consists of exactly a 1!a 2! a k! elements of. Observe further that there is a one-to-one correspondence between the set of k i=1 S a i -orbits on the set and the set of k-rectangles of the form (π 1,...,π k ), where each π i has a i blocks. Therefore, the number of such k-rectangles is { } n := a 1, a 2,..., a k n! a 1!a 2! a k!. Definition Let F 1 and F 2 be species of structures with F 1 [ ] = F 2 [ ] =. The arithmetic product of F 1 and F 2, denoted F 1 F 2, is defined by setting for each finite set U, (F 1 F 2 )[U] = (π 1 π 2 ) N [U] F 1 [π 1 ] F 2 [π 2 ], where the sum represents the disjoint union. In other words, an F 1 F 2 -structure on a finite set U is a tuple of the form ((π 1, f 1 ), (π 2, f 2 )), where f i is an F i -structure on the blocks of π i for each i. A bijection σ : U V sends a partition π of U to a partition π of V, namely, σ(π) = π = {σ(b) : B is a block of π}. Thus σ induces a bijection σ π : π π, sending each block of π to a block of π. The transport of structures for any bijection σ : U V is defined by (F 1 F 2 )[σ]((π 1, f 1 ), (π 2, f 2 )) = ((π 1, F 1 [σ π1 ](f 1 )), (π 2, F 2 [σ π2 ](f 2 ))). The arithmetic product on species satisfies the following properties. They are straightforward to check using the definition. 41

53 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES X X X X X X X F 1 X X F 2 F 1 X X X X X X F 2 Figure Arithmetic product F 1 F 2. Proposition (Maia and Méndez) Let F 1, F 2 and F 3 be species of structures with F i [ ] = for i = 1, 2, 3, and let X be the species of singleton sets. Then we have (commutativity) F 1 F 2 = F 2 F 1, (associativity) F 1 (F 2 F 3 ) = (F 1 F 2 ) F 3, (distributivity) F 1 (F 2 + F 3 ) = F 1 F 2 + F 1 F 3, (unit) F 1 X = X F 1 = F 1. X X F X X F X X X Figure X is the unit of the arithmetic product: F X = F. The following proposition about the arithmetic product of two molecular species could be taken as an alternative definition of the arithmetic product. Proposition Let X m /A and X n /B be two molecular species. That is, A is a subgroup of S m, and B is a subgroup of S n. Then X m A Xn B = Xmn A B, 42

54 CHAPTER 2. COMBINATORIAL THEORY OF SPECIES where A B is the product group acting on the set [mn] as defined by Definition Example Recall that the molecular species E 2 is the same as the species X 2 /S 2. The arithmetic product of the species E 2 with itself is a species concentrated on the cardinality 4. To be more precise, the set of E 2 E 2 -structures on a set U with U = 4 are obtained by taking the S 2 S 2 -orbits of 2 by 2 squares filled with elements of U, where the group S 2 S 2 acts by switching the rows and the columns. In other words, E 2 E 2 = X 4 /(S 2 S 2 ). E 2 E 2 a b c d Figure X 4 /(S 2 S 2 ). The species E 2 E 2 is isomorphic to the species We can quickly verify Proposition by observing that the X m /A-structures are A-orbits of linear orders on [m], and the X n /B-structures are B-orbits of linear orders on [n], and hence an (X m /A) (X n /B)-structure on [mn] is just a matrix whose rows and columns are A-orbits of linear orders on [m] and B-orbits of linear orders on [n], respectively. The automorphism group of such an (X m /A) (X n /B)-structure is therefore isomorphic to the product group A B. The associativity of the arithmetic product allows us to talk about the arithmetic product of a set of species. 43