ENUMERATION BY ALGEBRAIC COMBINATORICS. 1. Introduction

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2 How can we get from 64 down to 3 or 8? We might guess that it has something to do with symmetry. The hexagon has 2 symmetries: rotation by 0, 60, 20, 80, 240, or 300 degrees, three reflections through opposite vertices, and three reflections through opposite sides. There is no obvious relationship between the number of possibilities for a fixed necklace, the number of symmetries, and the number of distinct varieties, but surely there is one. In this paper, we use combinatorics and group theory to work through the problem of the six-beaded necklace and others like it. 2. Burnside s Theorem We begin by examining the number of possible n-colored necklaces and cubes under symmetry. We solve simple enumeration problems by observation, claim that the results hold for more complicated examples, then prove our claim algebraically. The result that we observe by counting cubes was popularized by William Burnside in 9, though he quotes it from a paper published by George Frobenius in 887 [3]. The theorem was also known to Augustin Cauchy in an obscure form by 845. [4] 2.. Basic Abstract Algebra. First, we give abstract algebra background necessary to describe an enumeration problem solvable by Burnside s theorem. Definition 2.. (Binary Operation [3]) Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. Definition 2.2. (Group [3]) Let G be a nonempty set together with a binary operation that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say that G is a group under this operation if the following three properties are satisfied. () Associativity. The operation is associative; that is, (ab)c = a(bc) for all a, b, c G. (2) Identity. There is an element e (called the identity) in G such that ae = ea = a for all a G. (3) Inverses. For each element a G, there is an element b G (called an inverse of a) such that ab = ba = e. Example 2.. A simple example of a group is the integers under addition modulo 4. In this case, Z 4 = {0,, 2, 3}. Addition mod 4 is a binary operation (the group is closed) since = 0, 0 + =, = 2, = 3, + = 2, + 2 = 3, + 3 = 0, = 0, =, and = 2. We need not check the operations in the other direction since addition is commutative. The group has an identity of 0, and each element has a unique inverse: 0 = 0, = 3, 2 = 2, and 3 =. 2

3 Beaded Necklace. We describe the symmetry of a 6-beaded necklace, or hexagon. Symmetry is the property that an object is invariant under certain transformations. The 6-beaded necklace is symmetric in 2 ways as shown in Figures 3, 4, and 5. The rotations of the necklace counterclockwise about its center by 0 o, 60 o, 20 o, 80 o, 240 o, or 300 o (the rotation by 0 o is the identity), denoted I, R 60, R 20, R 80, R 240, and R 300, Figure 3. The rotations of D 2. The reflections across the diagonal through each of the 3 pairs of opposite vertices, denoted D, D 2, and D 3, Figure 4. The reflections across the diagonals of D 2. The reflections across the side bisector for each of the 3 pairs of opposite edges, denoted E, E 2, and E Figure 5. The reflections across the side bisectors of D 2. 3

4 D 2 I R 60 R 20 R 80 R 240 R 300 D D 2 D 3 E E 2 E 3 I I R 60 R 20 R 80 R 240 R 300 D D 2 D 3 E E 2 E 3 R 60 R 60 R 20 R 80 R 240 R 300 I E 2 E 3 E D D 2 D 3 R 20 R 20 R 80 R 240 R 300 I R 60 D 2 D 3 D E 2 E 3 E R 80 R 80 R 240 R 300 I R 60 R 20 E 3 E E 2 D 2 D 3 D R 240 R 240 R 300 I R 60 R 20 R 80 D 3 D D 2 E 3 E E 2 R 300 R 300 I R 60 R 20 R 80 R 240 E E 2 E 3 D 3 D D 2 D D E D 3 E 3 D 2 E 2 I R 240 R 20 R 60 R 300 R 80 D 2 D 2 E 2 D E D 3 E 3 R 20 I R 240 R 80 R 60 R 300 D 3 D 3 E 3 D 2 E 2 D E R 240 R 20 I R 300 R 80 R 60 E E D 3 E 3 D 2 E 2 D R 300 R 80 R 60 I R 240 R 20 E 2 E 2 D E D 3 E 3 D 2 R 60 R 300 R 80 R 20 I R 240 E 3 E 3 D 2 E 2 D E D 3 R 80 R 60 R 300 R 240 R 20 I Table. The Cayley Table of D 2. Each entry in the table represents an operation in row composed with an operation in a column. We claim that these transformations form a group, called the dihedral group of order 2, denoted D 2. From the completeness of Table, the Cayley table of D 2, D 2 is closed under function composition. Notice that the element I is the identity of D 2 since every element composed with I is equal to itself. Also, every row of the table contains one I, so Theorem?? holds, or each element has an inverse. Since all of the properties of Definition 2.2 hold, the symmetric transformations of a 6-beaded necklace does indeed form a group. Next we examine the symmetric transformations on the numbered necklaces and notice that not every group operation affects every coloring of every necklace. For example, consider R 20. If we start with the pattern (23456) and apply R 20, we get (34562), then applying R 20 again, we get (56234). If we continue to rotate by 20 o, the pattern repeats. Therefore, this coloring remains the same under R 20 as long as beads (35) are the same color and beads (246) are the same color. If our necklace can contain two colors of beads, then we say that we have two choices of color for each set of fixed beads, so we have 2 2 possible arrangements fixed under R 20. The colorings fixed under each transformation are described in Table 2. Notice that the number of the total choices under all symmetries of two colored necklaces is 56. When we divide 56 by 2 (the number of symmetries), we get 3. By observation, there are 3 varieties of 2-colored necklaces that are different when the symmetries are applied: three choices where there are 3 beads of each color as shown in Figure 2 and ten more choices shown in Figure 6. In contrast, there are 2 6 = 64 possibilities for necklaces upon which we cannot apply a symmetric transformation Cube. The next question that we tackle is, given a cube that is painted dark on some faces and light on the others, keeping symmetries in mind, how many different possible varieties are there? To answer this 4

6 Light Gray Dark Gray Possibilities Varieties ( ) 0 = ( 5 6 ) = ( 6 ) 2 = 5 2 ( 6 ) 3 = 20 2 ( 6 ) 4 = 5 2 ( 6 ) 5 = 6 ( 6 ) 6 = Total Sum 64 0 Table 3. A comparison of the different possibilities and varieties of a 2-colored cube. Figure 7. The identity of D 24, also written as I or R 0. Figure 8. The rotations through the centers of the faces of D 24 : F, F 2, F 3. E 80,2,3,4,5,6 After that, there are rotations of 80 degrees about the six lines of symmetry that go from the center of one edge to the center of the diagonal edge. C 20,2,3,4, C 240,2,3,4 Finally, there are four rotations of 20 and 240 degrees about the lines of symmetry from a corner to the diagonal corner. Definition 2.3. (Multiplication Principle []) Let S be a set of ordered pairs (a, b) of objects, where the first object, a, comes from a set of size p and for each choice of object a, there are q choices for object b. Then the size of S is p times q: S = p q. 6

7 Figure 9. The rotations through the edges of D 24 : E, E 2, E 3, E 4, E 5, E 6. Figure 0. The rotations through the corner to corner diagonals of D 24 : C, C 2, C 3, C 4. There are 24 total unique symmetries, or possible orientations, of a cube. We can count these orientations in several ways. If we consider the cube as a unit cube in the first octant with a vertex at the origin, there 8 possible vertices that can rest at (0, 0, 0). For each vertex at the origin, there are 3 vertices that can be at (0, 0, ). By the multiplication principle, there are 24 orientations. Similarly, there are 6 faces that can rest on the xy-plane and for each such face there can be 4 faces on the yz-plane, so there are 24 symmetries as we expect. These symmetries form the dihedral group D 24 under the operation of function composition. Transformations φ in D 24 Possibilities Fixed (fix(φ)) Number ( fix(φ) ) I All 6 separately 2 6 = 64 F 90 left, right, 4 other sides 2 3 = 8 F2 90 front, back, 4 other sides 2 3 = 8 F3 90 top, bottom, 4 other sides 2 3 = 8 F 80 left, right, 2 opp. sides, 2 opp. sides 2 4 = 6 F2 80 front, back, 2 opp. sides, 2 opp. sides 2 4 = 6 F3 80 top, bottom, 2 opp. sides, 2 opp. sides 2 4 = 6 F 270 left, right, 4 other sides 2 3 = 8 F2 270 front, back, 4 other sides 2 3 = 8 F3 270 top, bottom, 4 other sides 2 3 = 8 E,2,3,4,5,6 80 top and bottom, front and left, back and right = 6 8 = 48 C,2,3, sides w/ common vertex, other 3 sides = 4 4 = 6 C,2,3, sides w/ common vertex, other 3 sides = 4 4 = 6 Total Sum 240 Table 4. The number of possibilities fixed under each operation in a cube, D 24. 7

8 Next, we will examine the number of sides fixed by each transformation. These are detailed in Table 4. We can see that for a cube, the sum over all elements of D 24 of the number of possible colorings fixed by each transformation is equal to the size of D 24 times the number of varieties. In this case, 240 = From this data and that from the 6-beaded necklace, we claim that the number of varieties is equal to the order of the dihedral group times the total number of choices of sets of sides fixed by each transformation The Proof of Burnside s Theorem. Now, we prove our conjecture after introducing more math. Definition 2.4. (Equivalence Relation [3]) An equivalence relation on a set S is a set R of ordered pairs of elements of S such that () (a, a) R for all a S (reflexive property) (2) (a, b) R implies (b, a) R (symmetric property) (3) (a, b) R and (b, c) R imply (a, c) R (transitive property) Definition 2.5. (Partition [3]) A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S. Theorem 2.. (Equivalence Classes Partition [3]) The equivalence classes of an equivalence relation on a set S constitute a partition of S. Conversely, for any partition P of S, there is an equivalence relation on S whose equivalence classes are the elements of P. Definition 2.6. (Subgroup [3]) If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G, denoted H G. Theorem 2.2. (Subgroup Test) Let G be a group and H a nonempty subset of G. Then, H is a subgroup of G if and only if ab H whenever a, b H, and a H whenever a H. Proof. If H is a subgroup of G, then H is nonempty since H contains the identity. Whenever a, b H, ab H and a H since H contains the identity of G and H is closed under multiplication. Conversely, if H is a nonempty subset of G such that whenever a, b H, ab H and a H we must show that H G. Let a be any element in H. Let a = b and deduce that e = aa H, so H contains the identity of G. Then, since H contains e and a, H contains the element ea = a H and H is closed under taking inverses. If a and b are any two elements of H, then H contains a and b, so H contains a(b ) = ab. Hence H is closed under multiplication, which proves H is a subgroup of G. Gallian 6, Dummit 48 8

9 Example 2.2. Consider the group Z 4 = {0,, 2, 3} and its subset {0, 2}. We will show that {0, 2} Z 4 using Theorem 2.2. First, we check for closure under addition: = 0, = 2, and = 0. Next, we check for closure under taking inverses: 0 = 0 and 2 = 2. Therefore {0, 2} is indeed a subgroup of Z 4. Definition 2.7. (Coset of H in G [3]) Let G be a group and H a subgroup of G. For any a G, the set {ah such that h H}, denoted by ah, is called the left coset of H in G containing a. Lemma (Properties of Cosets [3]) Let H be a subgroup of G, and let a and b belong to G. Then, () a ah, (2) ah = H if and only if a H, (3) ah = bh or ah bh =, (4) ah = bh if and only if a b H, (5) ah = bh if H is finite, (6) ah = Ha if and only if H = aha, (7) ah is a subgroup of G if and only if a H. Theorem 2.3. (Lagrange s Theorem [3]) If G is a finite group and H is a subgroup of G, then H divides G. Moreover, the number of distince left (right) cosets of H in G is G H. Proof. Let a H, a 2 H,..., a r H denote the distinct left cosets of H in G. Then, for each a G, we have ah = a i H for some i. By property of cosets, a ah. Thus, each member of G belongs to one of the cosets a i H. In symbols, G = a H... a r H. Now, property 3 of cosets shows that this union is disjoint, so G = a H + a 2 H a r H. Finally, since a i H = H for each i, we have G = r H. Definition 2.8. (Permutation of a Set S [3]) A permutation of a set S is a function from S to S that is both one-to-one and onto. Definition 2.9. (Permutation Group of S [3]) A permutation group of a set S is a set of permutations of S that forms a group under function composition. Definition 2.0. (Stabilizer of a Point [3]) Let G be a group of permutations of a set S. For each s S, let stab G (s) = {g G such that g(s) = s}. We call stab G (s) the stabilizer of s G. Example 2.3. The stabilizer of the cube in Figure 2.4 consists of I, F 90, F 80, and F 270. These are the rotations about the vertical line of symmetry through the middle of the top and bottom faces. 9

10 Figure. Stabilizer Example Definition 2.. (Orbit of a Point [3]) Let G be a group of permutations of a set S. For each s S, set orb G (s) = {g(s) such that g G}. The set orb G (s) is a subset of S called the orbit of s under G. Figure 2. Orbit Example Example 2.4. The orbit of the cube in Figure 2.4 consists of all orientations of a cube with one light side. Definition 2.2. (Elements Fixed by g [3]) For any group G of permutations on a set S and any g G, we let fix(g) = {s S such that g(s) = s}. This set is called the fix of g. Figure 3. Fix Example Example 2.5. The cubes fixed under C, corner-corner diagonal symmetry are pictured in Figure

11 Lemma ( Stabilizer is a Subgroup [3]) The stabilizer of s in G is a subgroup of G. Proof. Let s S and define stab G (s) = {g G such that g(s) = s}. The identity element of G, e, is in stab G (s) since e(s) = s. Let g, g 2 stab G (s). Then g g 2 (s) = g (g 2 (s)) = g (s) = s and stab G (s) is closed under function composition. Let g stab G (s) and consider g. Given that e(s) = s, g g(s) = g (s) = e(s) = s = g(s) = g g (s) so stab G (s) is closed under inverses. Lemma ( Orbits Partition [3]) Given that G is a group of permutations of a set S, the orbits of the members of S constitute a partition of S. Proof. To prove that the orbits of the members of S constitute a partition of S, we must show that the orbits are equivalence classes of the set S. Let s, s 2, s 3 S and g, g 2, g 3 G. Certainly, s orb G (s ). Now suppose s 3 orb G (s ) orb G (s 2 ). Then s 3 = g (s ) and s 3 = g 2 (s 2 ), and therefore (g 2 (g (s )) = s 2. So, if s 3 orb G (s 2 ), then s 3 = g 3 (s 2 ) = (g 3 (g 2 (g (s ))) for some g 3. This proves orb G (s 2 ) orb G (s ). By symmetry, orb G (s ) orb G (s 2 ). Theorem 2.4. (Orbit-Stabilizer Theorem [3]) Let G be a finite group of permutations of a set S. Then, for any s S, G = orb G (s) stab G (s). Proof. By Lagrange s Theorem (Theorem 2.3), G stab G (s) is the number of distinct left (right) cosets of stab G (s) in G. Thus, it suffices to establish a one-to-one correspondence between the left cosets of stab G (s) and the elements in the orbit of s. To do this, we define a correspondence T by mapping the coset g stab G (s) to g (s) under T. To show that T is a well-defined function, we must show that g stab G (s) = g 2 stab G (s) implies that g (s) = g 2 (s). But g stab G (s) = g 2 stab G (s) implies g g 2 stab G (s), so that (g g 2)(s) = s and, therefore, g 2 (i) = g (i). Reversing the argument from the last step to the first step shows that T is also one-to-one. We conclude the proof by showing that T is onto orb G (s). Let s 2 orb G (s ). Then g (s ) = s 2 for some g G and T (g stab G (s )) = g (s ) = s 2, so T is onto.

12 Theorem 2.5. (Burnside s Theorem [3]) If G is a finite group of permutations on a set S, then the number of distinct orbits of G on S is () G f ix(g). g G Proof. Let n be the number of pairs (g, s) with g G and s S such that g(s) = s. We can count the pairs in two ways. First, n = fix(g). Second, n = G (s). Therefore, g G s S stab = g G fix(g) stab G (s). s S Given that the orbits of the members of S constitute a partition of S (Lemma 2.3.2), if s and s 2 are in the same orbit, orb G (s ) = orb G (s 2 ) and stab G (s ) = stab G (s 2 ). Therefore, by the Orbit-Stabilizer Theorem (Theorem 2.4), stab G (s 2 ) = orb G (s ) stab G (s ) = G. s 2 orb G (s ) Finally, we can take the sum over all of the elements of G to get g G fix(g) = s S stab G (s) = G (number of orbits), and it follows that f ix(φ) = (number of distinct orbits). G φ G Example 2.6. (Necklace) Let S be the set of all possible color arrangements s, such that for a two-colored six-beaded necklace, S = 2 6 = 64. Let G be the dihedral group D 2. The group of symmetries g G which keep a particular coloring the same is stab G (s). The orb G (s) is the set of colorings s S that can be created by applying a symmetry g G. The set of colorings s S that remain fixed under a particular g G are in the set fix(g). The values for fix(g) can be found in Table 2. Therefore, according to Theorem 2.5, as we observe in Section 2.3. (number of varieties) = G g G fix(g) = 56 = 3 2 Example 2.7. (Cube) Let S be the set of all possible color arrangements s, such that for a two-colored cube, S = 2 6 = 64. Let the permutation group G be D 24. The subgroup of G which keep a particular coloring the same is stab G (s). The set of colorings in S that can be created from a particular s S by 2

13 applying the permutation g G is orb G (s). The fix(g) is the set of colorings in S that are invariant under g G. According to Burnside s theorem (Theorem 2.5), The number of orbits of G on S, or the number of colorings distinct under G is 24 f ix(g). g G The values for fix(g) can be found in Table 4, so g G orbits, or varieties, for a two-colored cube. fix(g) = 240. The result is that there are 0 distinct 2.5. Tetrahedron. Burnside s theorem does not place a restriction on the number of color choices. For example, examine a tetrahedron colored with k colors using Burnside s theorem (Theorem 2.5). A tetrahedron has four faces so our set S = {bottom, left, right, front} and S = 4. Let their be k possible colors for each face. Therefore, we count k 4 possible colorings for an immobile tetrahedron. A tetrahedron has 2 different symmetries, so G = 2 and the group G contains the following transformations. I The first is the identity operation, or any rotation of zero degrees. V 20,2,3,4, V 240,2,3,4 Next are rotations of 20 and 240 degrees about the 4 lines of symmetry from each vertex to the center of the opposite side. E 80,2,3 Finally, there are rotations of 80 degrees about the three lines of symmetry that go from the center of one edge to the center of the center of the opposite edge. The fix(g) for each g G are represented in Table 5. Substituting these numbers into Theorem 2.5, we get 2 (k4 + k 2 ) = (number of distinct k-colored tetrahedrons under G). If we substitute in k = 2 to calculate the number of 2-colored tetrahedrons, we get 2 ( ) = 60 = 5. 2 We can verify that there are indeed five varieties of two-colored tetrahedrons by observation. 3. Pólya s Theorem 3.. Motivation. In the cube counting problem (Example 2.7), Burnside s theorem (Theorem 2.5) allows us to calculate the number of varieties of two-colored cubes under the group D 24, but it does not tell us much about those varieties. Let the colors of the cube be denoted L for the light colored faces and D for 3

14 g f ix(g) f ix(g) I All 4 separately k 4 V 20,2,3,4 bottom, other three sides 4 k 2 V,2,3,4 240 bottom, other three sides 4 k 2 E,2,3 80 top and left, bottom and right 3 k 2 Total Sum k 4 + k 2 Table 5. The number of possible colorings fixed under each symmetry of a k-colored tetrahedron. the dark colored faces. We generate a polynomial for some g G such that each grouping of sides in fix(g) is represented by the factor (L a + D a ) where a is the number of sides in the grouping. For example, the identity element of G is represented by the polynomial (L + D) 6 since each face can be colored separately and remain invariant under I for each of the 6 sides that complete our cube. Next, consider F In this case, the top of the cube can be any color, the bottom can be any color, the front and back must be the same color, and the left and right sides must be the same color. The top and the bottom are both represented by (L+D). The set containing the front and back is represented by (L 2 +D 2 ). We take the product of all four terms, so our polynomial representing F 80 3 is (L + D) 2 (L 2 + D 2 ) 2. The polynomials for the other terms are represented in Table 6. If we take the sum of all of the polynomials (each polynomial represents an element of G) for a two-colored cube, we get (2) 24(L 6 + L 5 D + 2L 4 D 2 + 2L 3 D 3 + 2L 2 D 4 + LD 5 + D 6 ). Notice that in Equation 2 that after factoring out G (24 in this case), the coefficients of the terms are informative. The coefficient of LD 5 = and there is exactly variety with light side and 5 dark sides. The coefficient of L 3 D 3 = 2 and there are exactly two varieties of cubes with 3 light sides and 3 dark sides. We did not start with any more information than we did for Burnside s theorem (Theorem 2.5), but we learn more about the colorings using polynomials. φ D 24 Polynomial Expansion I (L + D) 6 L 6 + 6L 5 D + 5L 4 D L 3 D 3 + 5L 2 D 4 + 6LD 5 + D 6 F 90 F 80 F 270 E 80 C 20 C 240,2,3 3 (L + D) 2 (L 4 + W 4 ) 3 (L 6 + 2L 5 D + L 4 D 2 + L 2 D 4 + 2LD 5 + D 6 ),2,3 3 (L + D) 2 (L 2 + D 2 ) 2 3 (L 6 + 2L 5 D + 3L 4 D 2 + 4L 3 D 3 + 3L 2 D 4 + 2LD 5 + D 6 ),2,3 3 (L + D) 2 (L 4 + W 4 ) 3 (L 6 + 2L 5 D + L 4 D 2 + L 2 D 4 + 2LD 5 + D 6 ),2,3,4,5,6 6 (L 2 + D 2 ) 3 6 (L 6 + 3L 4 D 2 + 3L 2 D 4 + D 6 ),2,3,4 4 (L 3 + D 3 ) 2 4 (L 6 + 2L 3 D 3 + D 6 ),2,3,4 4 (L 3 + D 3 ) 2 4 (L 6 + 2L 3 D 3 + D 6 ) Sum 24(L 6 + L 5 D + 2L 4 D 2 + 2L 3 D 3 + 2L 2 D 4 + LD 5 + D 6 ) Table 6. The polynomials for each symmetry of a 2-colored cube in D 24. 4

15 3.2. Cycle Index of a Permutation Group. We begin our build up to Pólya s theorem with more discussion of permutations. The cycle index of a permutation group is a polynomial used in Pólya s theorem and many of its extensions. Theorem 3.. ( Products of Disjoint Cycles [3]) Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles. Proof. Let S be a finite set where S = m and let g be a permutation of S. To write g in disjoint cycle form, choose any s S and let s 2 = g(s ), s 3 = g(g(s )) = g 2 (s ),... until s = π l (s ) for some l. We know that such an l exists because the sequence must be finite. Say g i (s ) = g j (s ) for some i and j with i < j. Then s = g l (s ), where l = j i. We express this relationship among s, s 2,..., s l as g = (s, s 2,..., s l )... It is possible that we have not exhausted the set S in this process. So, we pick any t S not appearing in the first cycle and generate a new cycle. That is, we let t 2 = g(t ) and so on until we reach t = g k (t ) for some k. This new cycle has no elements in common with the previously constructed cycle. For, if it did, then g i (s ) = g j (t ) for some i and j. But then, g i j (s ) = t, and therefore t = s r for some r. This is a contradiction since t is not contained in cycle s. Continuing this process until we run out of elements of S, our permutation will appear as g = (s, s 2,..., s l )(t,..., t k )...(c,..., c r ). In this way, every permutation can be written as a product of disjoint cycles. If g is a permutation of a finite set S, we can split S into disjoint cycles, or subsets of S cyclically permuted by g. Given a cycle of length l, the elements of the cycle are s, g(s), g 2 (s),..., g l (s). Definition 3.. (Type [2]) If permutation g splits S into b cycles of length, b 2 cycles of length 2, etc., g is of type {b, b 2, b 3,... }. We can see that b i = 0 for i > S. Also, b i = 0 for all but at most a finite number of i s. Also, S = b + 2b 2 + 3b Definition 3.2. (Cycle Index [2]) If G is a permutation group, for each g G, consider the product x b x 2 b2 x m b m where g is of type {b, b 2,...}. The cycle index of G is the polynomial (3) P G (x, x 2,..., x m ) = G b x b2 b x 2 x m m Example 3.. Let G = {e}, or the identity permutation of S. We can see that e is of type {m, 0, 0,...} g G since it has m cycles of length, so its cycle index is equal to x m. 5

16 Example 3.2. Let G be the permutation group generated by (23)(456)(78). The each element of group G is of type {0,, 2} and therefore P G (x, x 2, x 3 ) = x 2 x 2 3 = x 2 x Example 3.3. [2]Let S be the set of cube faces and G be the permutation group of S. Using our previously defined notation and descriptions of the symmetries of a cube, we can visualize I is of type {6, 0, 0,...}. F 80,2,3 are of type {2, 2, 0, 0,...}. F 90,270,2,3 are of type {2, 0, 0,, 0,...}. E 80,2,3,4,5,6 are of type {0, 3, 0, 0,...}. V 20,240,2,3,4 are of type {0, 0, 2, 0, 0,...}. Since G = 24 and = 24, our list of symmetries is exhaustive. We calculate the cycle index of G to be (4) P G = 24 (x6 + 3x 2 x x 2 x 4 + 6x x 2 3). g G The following theorems and definitions help us understand the next example: the cycle index of the Cayley representation of a general finite set. Definition 3.3. (Group Isomorphism) An isomorphism φ from a group G to a group Ḡ is a one-to-one and onto mapping from G to Ḡ that preserves the group operation. That is, (5) φ(g g 2 ) = φ(g )φ(g 2 ) for all g, g 2 G. Theorem 3.2. (Cayley s Theorem [3]) Every group is isomorphic to a group of permutations. Proof. Let G be any group. We will find a group Ḡ of permutations that is isomorphic to G. For any g G, define function T g : G G by T g (x) = gx for all x G. To prove that T g is a permutation on the set of elements in G, we show that T g : G G is one-to-one and onto. For all g G, T g (x) = gx G by the group property of closure. Given gx = gx 2, by cancellation x = x 2, and thus T g is one-to-one. Given any x G, there exists an x 2 G such that T g (x 2 ) = gx 2 = x thus T g is onto. For the permutation T g, we show that that if g runs through G, then the T g s form a permutation group Ḡ of G. First, Ḡ is nonempty since if e is the identity of G. For all x G, x = ex = g e (x) Ḡ. For all g, g 2, g 3 G, Ḡ is associative since (T g T g2 )T g3 (x) = (g g 2 )g 3 x = g (g 2 g 3 )(x) = T g (T g2 T g3 )x. Next we 6

17 show that Ḡ is closed under multiplication and taking inverses. If g and g 2 are in group G, g g 2 is also in G. From the way we defined T g, we see that T g T g2 (x) = g g 2 x = T gg 2 (x) Ḡ. The inverse of T g = gx is Tg (x) = g x since T g Tg (x) = gg x = ex = x for all x G and therefore all T g Ḡ. By Theorem 2.2, Ḡ is a group. For every g G, define φ(g) = T g. If T g = T h, then T g (e) = T h (e) or ge = he. Thus, g = h and φ is one-to-one. By the way Ḡ is constructed, φ is onto. For some g, g 2 G, φ(g g 2 ) = T g T g2 = φ(g )φ(g 2 ). Therefore φ is an isomorphism from G to Ḡ. Ḡ is called the Cayley representation of the group G. Example 3.4. [2]Let S be any finite group, and let S = m. If s is a fixed element of S, then for all x S, the mapping S sx is a permutation of S. Denoting this permutation g s, if s runs through S, the g s s form a permutation group G. By Cayley s Theorem and its proof, G is isomorphic to S and is the Cayley representation of S. We are interested in its cycle index. If s S, let the order of s be s = k(s). Permutation g s splits S into cycles of length k(s), since if s S, the cycle obtained by g s is {x, sx, s 2 x,..., s k(s) x = x}. These cycles are cosets of S. It follows from Lagrange s Theorem (Theorem 2.3) that for each k(s) m and there are the cycle index of G is (6) P G = [x k(s) ] m m s S k(s). m k(s) cycles of length k(s). Thus, 3.3. Weighted Form of Burnside s Theorem. Earlier, we discussed Burnside s theorem in which the elements of G are permutations. It is a trivial extension so say that now the elements of G act as permutations. That is, to each g G we have attached a permutation of S denoted π g. The mapping is homomorphic, or π gg = π g π g for all g, g G. We say that two elements of S are equivalent, or s s 2, if there exists a g G such that π g s = s 2. The equivalent classes of S are called transitive sets. We now state our weighted form of Burnside s theorem. Theorem 3.3. (Weighted Burnside s Theorem [2]) Given set S, group G, and permutation group π G, the number of transitive sets equals (7) G ψ(g), g G where G denotes the number of elements of G, and for each g G, ψ(g) denotes the number of elements of S that are invariant under π g. 7

18 This is the weighted version of Burnside s theorem since the different elements of G need not correspond to different permutations π. Since we are adding up all of the π g s, some may be counted more than once Functions and Patterns. Let D, domain, and R, range, be finite sets. Let the set of all functions from D into R be denoted R D, where the number of elements in R D is R D since for each d D, we have R choices for f(d). Let G be a permutation group of D. Definition 3.4. (Pattern [2]) Given f, f 2 R D, we call f and f 2 equivalent, or f f 2 if there exists a g G such that f (gd) = f 2 (d) for all d D. These equivalence classes are called patterns. We confirm that f g = f 2 is indeed an equivalence relation. The relation is reflexive because f f by the identity permutation in G. The relation is symmetric since f f 2 then f 2 f since if g G implies g G. The relation is transitive since if f f 2 and f 2 f 3, then f f 3 because G is closed under multiplication. Example 3.5. [2]Let D be the set of six faces of a cube. Let G be the group of permutations of D by symmetry. Let R be the colors light and dark. The set of all possible colorings of a fixed cube is R D and the number of colorings is R D = 2 6. The number of varieties of rotatable cubes, also called transitive sets, orbits of G, or patterns, is 0. The method for arriving at these numbers has been explained in Example 2.7. Example 3.6. [2]Let D = {, 2, 3}, let G be the symmetric group of D (the group of all permutation), and let R = {x, y}. We can see that there are eight functions, but only four patterns as shown in Table 7. Since Pattern Corresponding Functions x 3 f() = f(2) = f(3) = x x 2 y f() = f(2) = x, f(3) = y; f() = f(3) = x, f(2) = y; f(2) = f(3) = x, f() = y y 2 x f() = f(2) = y, f(3) = x; f() = f(3) = y, f(2) = x; f(2) = f(3) = y, f() = x y 3 f() = f(2) = f(3) = y Table 7. The functions and patterns when D = {, 2, 3}, R = {x, y}, and G = S 3. the symmetric group does not depend on the order of the factors, functions f and f 2 are equivalent if and only if the products f ()f (2)f (3) and f 2 ()f 2 (2)f 2 (3) are identical. We have stated the weighted version of Burnside s theorem. Now, we define the weights Weights of Functions and Patterns. Definition 3.5. (Ring [3]) A ring R is a nonempty set with two binary operations, addition and multiplication, such that for all a, b, c R: 8

19 () a + b = b + a. (2) (a + b) + c = a + (b + c). (3) There is an additive identity 0. That is, there is an element 0 in R such that a + 0 = a for all a R. (4) There is an element a R such that a + ( a) = 0. (5) a(bc) = (ab)c. (6) a(b + c) = ab + ac and (b + c)a = ba + bc. A ring is called a commutative ring if ab = ba for all a, b R. Definition 3.6. (Weight [2]) Given that D and R are finite sets, the set of all functions from D into R is R D, and G is a permutation group of D. We assign each element in R a weight. This weight can be a number, a variable, or generally, an element of a commutative ring containing the rational numbers. The weight for each r R is denoted w(r). Definition 3.7. (Weight of a Function [2]) The weight of a function f R D, denoted W(f) is (8) W (f) = d D w[f(d)]. If f f 2, or the functions belong to the same pattern, then they have the same weight. This is because if f g = f 2 for all g G. Using the fact that the same factors are involved and they are commutative we conclude that d D w[f (d)] = d D w[f (g(d))] = d D w[f 2 (d)]. Definition 3.8. (Weight of a Pattern [2]) The weight of a pattern F, denoted W (F ), is equal to the weight of any (and every) function in F. Given any f F, (9) W (F ) = d D w[f(d)]. Example 3.7. [2]Recall the light gray and dark gray cube from Example 3.5. Let D be the set of faces, R = {light, dark}, and let G be is the set of symmetries. We set weights, w(light) = x and w(dark) = y, the weights for each of the 0 patterns are: x 6, x 5 y, x 4 y 2, x 3 y 3, x 3, y 3, x 2 y 4, xy 5, y 6. Different patterns need not have different weights. For example, x 3 y 3 represents the pattern with three light sides and three dark sides in two ways: three dark sides meet at a single corner, three dark sides that wrap around the cube. Example 3.8. [2]Recall Example 3.6 where D = {, 2, 3}, G is the symmetric group of D, and R = {x, y}. We set w(x) = x and w(y) = y so x 3, x 2 y, xy 2, and y 3 become the pattern weights. In this case, different 9

20 Figure 4. Two cubes with the same weight and different patterns. patterns have different weights. If we set w(r) = for all r R, then W (f) = for all f F, and therefore W (F ) = for each pattern Store and Inventory. Definition 3.9. (Store [2]) Given that D and R are finite sets and every r R has a weight w(r), the set of all functions from D into R is R D, and G is a permutation group of D. We call R the store because it is the set from which we choose function values. Definition 3.0. (Inventory [2]) Since the values of w(r) must come from a commutative ring, the weights can be added to each other. The store enumerator, or inventory of R is (0) inventory of R = r R w(r). Example 3.9. Let us say that a cafe owner has 3 candy bars that cost \$ each, 2 bottles of pop that cost \$3 each, and 4 hamburgers in stock that cost \$5 each. His store is R = {c, c 2, c 3, p, p 2, h, h 2, h 3, h 4 }. Depending on the values to which we set the weights, the inventory of R will give us different information: To get complete information about the store, we can set the weight values at w(c ) = c, w(c 2 ) = c 2,..., w(h 4 ) = h 4. Then the inventory of R = c + c 2 + c 3 + p + p 2 + h + h 2 + h 3 + h 4. We might only be interested in the number of items of each type. In this case, we set the weights as w(c ) = w(c 2 ) = w(c 3 ) = c and w(p ) = w(p 2 ) = p and w(h ) = w(h 2 ) = w(h 3 ) = w(h 4 ) = h. In this case, the inventory of R = 3c + 2p + 4h. To get the value of the store, we set the weights as w(c ) = w(c 2 ) = w(c 3 ) = and w(p ) = w(p 2 ) = 3 and w(h ) = w(h 2 ) = w(h 3 ) = w(h 4 ) = 5. In this case, the inventory of R = = \$29. To get the number of items in the store, we set the weights as w(c ) =, w(c 2 ) =,..., w(h 4 ) =. Then the inventory of R = = 9. 20

21 3.7. Inventory of a Function. We are given that D and R are finite sets, every r R has a weight w(r), the set of all functions from D into R is R D, every function f R D has weight w(f) = d D w[f(d)], and G is a permutation group of D. The inventory of R D is the D th power of the inventory of R: () inventory of R D = f [ D W (f) = w(r)]. r R We can write the D th power of w(r) as the product of the D factors. In each factor r R w(r), r R pick a term w(r). By taking the product of these terms, we get one term of the full expansion. Since each factor has R terms, there are R D terms in the full expansion. Next, we take f(d) = r which allows us to say that the selection of a term from a factor is a one-to-one mapping f of D into R. To every f there corresponds the term (2) w[f(d)] d D of the full expansion. Since the term in (2) is exactly the definition of W (f) in (8), or the weight of a function, the full product is equal to the sum of all W (f), which is the inventory of R D. [2]Next, we will derive the inventory of S, a certain subset of R D. First, partition D into k disjoint components D, D 2,...D k such that D = D + D D k. Let S be the set of all functions f that are constant on each component. They may, but do not have to be different on different components. Let f be expressed as a composition of two functions, φ and ψ, such that f = φψ. The function ψ maps d onto the index of the component to which d belongs, so we always have d D ψ(d). The function φ is the mapping from {, 2,..., k} into R, so there are R k possibilities for φ. Therefore, k (3) inventory of S = [w(r)] Di. i= r R We can see this same result for the inventory of S by examining the full expansion of the product. To select one term of the product, we pick a value for φ which is a mapping from {, 2,..., k} into R. This φ produces 2

22 the term {w[φ()]} D {w[φ(2)]} D2 {w[φ(k)]} Dk = If φψ = f, then this term is exactly W (f) since and k {w[φ(i)]} Di = w{φ[ψ(d)]}, d D i i= d D i w[f(d)] = W (f). k {w[φ(i)]} Di. In this way, each f S is obtained exactly once. Thus the sum of W (f) for all f S is equal to the sum of all terms of the expansion of the inventory of S. i= Example 3.0. [2]Given m identical counters, we pass them out between three people, P, P 2, P 3, under the constraint that you give the same number to P and P 2. How many ways are there to do this? Let D = {P, P 2, P 3 }, R = {0,, 2,...}, f(p ) = f(p 2 ), and f(p ) + f(p 2 ) + f(p 3 ) = m. First we separate D into D = {P, P 2 } and D 2 = {P 3 }. Then we set the weights of R such that 0,, 2, 3,... have weights, x, x 2, x 3,... and the weight of the functions W (f) = x m. If S is the set of all functions constant on each D i, and by (3), the inventory of the set of all functions constant on each D i equals (4) ( + x 2 + x 4 + )( + x + x 2 + ). We are seeking the coefficient of x m in the expansion (4). Using the formula for the sum of a geometric series, we can simplify our inventory to Then we expand this result using partial fractions to x = x i, i=0 x 2 x. x 2 x = A + x + B ( x) 2 + C x = A( x)2 ( x) + B( + x)( x) + C( + x)( x) 2 ( + x)( x) 2 ( x) 22

23 Next, substitute, A = 4, B = 2, and C = 4 to get 4 ( x)2 ( x) ( + x)( x) 2 ( x) + 2 ( + x)( x) ( + x)( x) 2 ( x) + 2 ( + x)( x)2 ( + x)( x) 2 ( x) For the number of possible distributions we obtain = 4 ( + x) + 2 ( x) ( x). (5) 2 (m + ) + 4 [ + ( )m ] so the number of function is 2 m + if m is even and 2 m + 2 if m is odd. Let [x k ] denote the coefficient of the x k -term. We will verify (5) result directly for small values of m: If m =, the inventory is ( + x 2 )( + x), and the x-term is x so [x] =. In words, there is possibility for the distribution of counter and that is 0 to P and P 2 and to P 3. If m = 2, the inventory is ( + x 2 + x 4 )( + x + x 2 ), and the x 2 -term is x 2 + x 2 so [x 2 ] = 2. In words, there are 2 possibilities for the distribution of 2 counters and those are: each to P and P 2 and 0 to P 3 ; 0 to P and P 2 and 2 to P 3. If m = 3, the inventory is ( + x 2 + x 4 + x 6 )( + x + x 2 + x 3 ), and the x 3 -term is x 3 + x 2 x so [x 3 ] = 2. In words, there are 2 possibilities for the distribution of 3 counters: each to P, P 2, and P 3 ; 0 to P and P 2 and 3 to P 3. If m = 4, the inventory is ( + x 2 + x 4 + x 6 + x 8 )( + x + x 2 + x 3 + x 4 ), and the x 4 -term is x 4 + x 2 x 2 + x 4 so [x 4 ] = 3. The possibilities for the distribution of 4 counters are: 2 each to P and P 2 and 0 to P 3 ; each to P and P 2 and 2 to P 3 ; 0 each to P and P 2 and 4 to P 3. These results are concur with (5) and also give us more information on the possible distributions The Pattern Inventory. Theorem 3.4. (Pólya s Fundamental Theorem [2]) We are given finite sets D and R, a permutation group G of D, each r R has weight w(r), each function f R D and the patterns F have weights W (f) and W (F ). The pattern inventory is { (6) W (F ) = P G w(r), [w(r)] 2, } F r R r R r R[w(r)] 3,... where P G is the cycle index of G. Proof. Let R and D be finite sets, R D be the set of all functions from D into R, and G a permutation group of D. Let S be the subset of function f R D such that W (f(d)) = w. If g G, and f (d) = f 2 (gd), then f 23

24 and f 2 belong to the same pattern and have the same weight. Therefore, if f S, then f g S. Thus for each g G there is a mapping π g of S into itself, π g (f(d)) = f(g d). This mapping is a permutation since (π g ) = π g ( ). For each f S, if g, g G, for all d D, π gg (f(d)) = f((gg ) d) = f(g g d) = π g (f(g d)) = π g (π g f(d)) and the mapping g π g is homomorphic. We say the f and f 2 are equivalent if and only if f (d) = f 2 (g d) = π g (f 2 (d)). Therefore, the patterns contained in S are equivalence classes. According to the weighted Burnside s theorem, the number of patterns contained in S is equal to G ψ w (g), g G where ψ w (g) is the number of functions f such that W (f) = w. This is equivalent to f(g (d)) = f(d) = π g (f(d)) or f(d) = f(g(d)). The patterns contained in S all have weight w; therefore if we multiply (7), the weighted Burnside theorem, by w and sum over all possible values of w, we obtain the pattern inventory W (F ) = G w ψ w (g). w g G We have (g) w ψ w (g) = W (f), w f where (g) f means the summation over all f R D that satisfies f(d) = f(gd) for all d D. It follows that W (F ) = G W (f). g G (g) f 24

25 (g) In order to evaluate W (f), we remark that g is a permutation of D, and therefore splits D into cycles f that are cyclically permuted by g. The condition f(d) = f(gd) means that f(d) = f(gd) = f(g 2 d) =, where f is constant on each cycle of D. Conversely, each f that is constant on every cycle automatically satisfies f(gd) = f(d) since g(d) always belongs to the same cycle as d itself. (g) D, D 2,..., D k, then the sum is just the inventory of S from (3). f Thus if the cycles are Let {b, b 2,...} be the type of g. This means, among the numbers D,..., D k, the number occurs b times, the number 2 occurs b 2 times, etc. Consequently, we have (g) f { b { } b2 = w(r)} [w(r)] 2. r R The number of factors in finite, but we need not invent a notation for the last one since all b i are zero from a certain i onward. The expression above can be obtained by the substitution of r R x = r R w(r), x 2 = r R[w(r)] 2, x 3 = r R[w(r)] 3,... into the product x b xb2 2 xb3 3, which is the term corresponding for g in G P G. Summing with respect to g and dividing by G, we infer that the value of the pattern index, W (F ), is obtained by making the same substitution into P G (x, x 2, x 3,...), and this proves Pólya s theorem. Example 3.. If all of the weights are chosen to be equal to unity, then we obtain that the number of patterns is equal to P g ( R, R, R,...). We can show this by directly substituting w(r) = for all r R. The pattern inventory becomes simply the number of pattern which is equal to P G ( R, R, R,...). Example 3.2. Consider the light and dark painted cube. Let D bet he set of faces of the cube, G be the set of rotational symmetries, R be the set of colors, {light, dark}. Let the weights w(light) = and w(dark) =. Then from Example 3., the number of patterns = P G ( R, R, R,..) and we obtain 24 ( ) = 0. This is the same number of patterns that we discovered by inspection and then derived by Burnside s theorem. 25

26 Example 3.3. [2]Now, let the weights w(light) = x and w(dark) = y. First, calculate the pattern inventory using Pólya s theorem, Recall the the cycle index of G is { W (F ) = P G w(r), [w(r)] 2, } F r R r R r R[w(r)] 3,.... P G (x, x 2, x 3, x 4 ) = 24 (x6 + 3x 2 x x 2 x 4 + 6x x 2 3). Next, the we substitute r R[w(r)] i for the x i s. Therefore, the pattern inventory is 24 [(x + y)6 + 3(x + y) 2 (x 2 + y 2 ) 2 + 6(x + y) 2 (x 4 + y 4 ) + 6(x 2 + y 2 ) 3 +8(x 3 + y 3 ) 2 ]. Expanding the pattern inventory, we get 24 (x6 + x 5 y + 2x 3 y 3 + 2x 2 y 4 + xy 5 + y 6 ). This polynomial can be interpreted to give us complete information about the possible colorings of the cube. For example, we might ask how many patterns have four light faces and two dark faces. The answer is the coefficient of the x 4 y 2 -term, [x 4 y 2 ] = 2. This is consistent with the value that we found in (2). Also, the sum of the coefficients is predictably equal to ten, which is the total number of patterns. Example 3.4. [2]Let D be a finite set, and let G be a permutation group of D. We call two subsets of D, D and D 2, equivalent if for some g G, we have gd = D 2. In other words, D 2 is the set of all elements gd obtained by letting d run through D. We can form classes of equivalent subsets in this manner. We can create functions, f g (d) = gd, that are in one-to-one correspondence with the subsets. Let R = {light, dark}, w(light) =, and w(dark) =. In this case, the number of subset classes is equal to the pattern inventory, which by Pólya s theorem, is equal to P G (2, 2, 2,...). Now let w(light) = w and w(dark) =, where w is a variable. Then the subsets of k elements correspond to functions f with W (f) = w k. Therefore, the number of equivalence classes which consist of subsets of k elements is equal to the coefficient of w k in the pattern inventory, P G ( + w, + w 2, + w 3,..). Summing over all k, we obtain P G (2, 2, 2,...) since the sum of the coefficients of a polynomial p(w) is equal to p(). This example relates strongly to the cube problem. For example, let a subset D be equal to the set of all cubes with just one side light. There exists some g G such that all cubes with one light side are equivalent to all of the other cubes with one light side, but not to any cubes with a different number of light sides. 26

27 Then the cycle index is the same as it is for any two colored cube. The pattern inventory is now 24 [(w + )6 + 3(w + ) 2 (w 2 + ) 2 + 6(w + ) 2 (w 4 + ) + 6(w 2 + ) 3 + 8(w 3 + ) 2 ]. If we expand the pattern inventory, we get w 6 + w 5 + 2w 3 + 2w 2 + w +. We can create the same two colored cubes as before using these simplified calculations simply by painting the complementary faces dark. 4. Generalizations and Extensions of Pólya s Theorem 4.. Generalization of Pólya s Theorem. In Pólya s theorem, G, a permutation group of D induced an equivalence relation on the mappings f R D. Now, we consider a second permutation group H of R with a new equivalence relation on the mappings f R D which is defined based on both groups. Let f be equivalent to f 2 if there exists a g G, h H such that f g = hf 2. We now show that (7) f g = hf 2 or f (g(d)) = h(f 2 (d)) for all d D is an equivalence relation. () If g and h are identity permutations, f (g(d)) = h(f (d)) so f f and (7) satisfies the reflexive property. (2) If f f 2, there exists a g G, h H such that f (g(d)) = h(f 2 (d)) for all d D. Since g (d) D since G is a permutation group, f (g(g (d))) = h(f 2 (g (d)), so f 2 (g (d)) = h (f (d)) for all d in D so f 2 f. Therefore (7) satisfies the symmetric property. (3) If f f 2 and f 2 f 3, then there exists g, g G, h, h H such that f (g(d)) = h(f 2 (d )) and f 2 (g (d)) = h (f 3 (d)). Let d = g (d). Then f (g(g (d)) = h(f 2 (g (d))) = h(h (f 3 (d))) since g, g G and h, h H. Therefore f f 3 and (7) satisfies the transitive property. Therefore (7) is an equivalence relation so R D splits into equivalence classes, or patterns. We assume that each f R D has a weight W (f) such that the the weights are elements of a commutative ring. We need not assume that the W (f) are obtained from the elements of R, but we do assume that equivalent functions have the same weight, or (8) f f 2 = W (f ) = W (f 2 ). 27

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