9.4 Applications to Combinatorial Counting Problems

Polya's Enumeration Theorem is a game-changer for counting problems with symmetries. It uses cycle index polynomials to find distinct colorings of objects, like necklaces or chemical compounds, considering their symmetries.

This theorem shines in real-world applications. From designing unique circular logos to counting distinct isomers in chemistry, Polya's Theorem helps solve complex counting problems by factoring in rotational and reflectional symmetries.

Counting Problems with Polya's Theorem

Overview of Polya's Enumeration Theorem

• Polya's Enumeration Theorem is a powerful tool for solving counting problems involving symmetries and equivalence classes
• The theorem states that the number of distinct colorings of a set of objects, up to permutations of the colors, is given by the cycle index polynomial of the permutation group acting on the objects
• The cycle index polynomial is a generating function that encodes information about the cycle structure of the permutations in the group
• To apply Polya's Theorem, one must identify the set of objects to be colored (beads in a necklace), the permutation group acting on the objects (rotations), and the number of available colors

Real-world Applications of Polya's Theorem

• Real-world counting problems that can be modeled using Polya's Theorem include counting the number of distinct necklaces, bracelets, or chemical compounds with certain symmetries
• Polya's Theorem can be used to count the number of unique designs for a circular logo with a fixed number of sections and colors
• The theorem is also applicable in chemistry to enumerate the number of distinct isomers of a compound, considering rotational and reflectional symmetries of the molecule

Necklace and Bracelet Counting

Necklace Counting with Polya's Theorem

• A necklace is a circular arrangement of beads, where two necklaces are considered equivalent if one can be obtained from the other by rotation
• The permutation group acting on a necklace is the cyclic group, which consists of rotations of the necklace
• To count the number of distinct necklaces with a given number of beads and colors, one must compute the cycle index polynomial of the cyclic group and substitute the number of colors into the polynomial
• For example, to count the number of distinct necklaces with 4 beads and 2 colors, the cycle index polynomial is $\frac{1}{4}(x_1^4 + 2x_2^2 + x_4)$, and substituting 2 for each variable yields 6 distinct necklaces

Bracelet Counting with Polya's Theorem

• A bracelet is similar to a necklace, but two bracelets are considered equivalent if one can be obtained from the other by rotation or reflection
• The permutation group acting on a bracelet is the dihedral group, which consists of rotations and reflections of the bracelet
• To count the number of distinct bracelets with a given number of beads and colors, one must compute the cycle index polynomial of the dihedral group and substitute the number of colors into the polynomial
• For instance, to count the number of distinct bracelets with 4 beads and 2 colors, the cycle index polynomial is $\frac{1}{8}(x_1^4 + 3x_2^2 + 2x_1^2x_2 + 2x_4)$, and substituting 2 for each variable yields 4 distinct bracelets

Non-Isomorphic Graph Counting

Applying Polya's Theorem to Graph Isomorphism

• Two graphs are considered isomorphic if there exists a bijection between their vertex sets that preserves adjacency
• The permutation group acting on the vertices of a graph is the symmetric group
• To apply Polya's Theorem, one must consider the cycle structure of the permutations in the symmetric group and the number of ways to assign edges to each cycle type
• The resulting cycle index polynomial can be used to compute the number of non-isomorphic graphs by substituting the number of available edge configurations into the polynomial

Example of Non-Isomorphic Graph Counting

• To count the number of non-isomorphic graphs with 3 vertices, one must consider the cycle index polynomial of the symmetric group $S_3$, which is $\frac{1}{6}(x_1^3 + 3x_1x_2 + 2x_3)$
• For each term in the polynomial, one must determine the number of ways to assign edges to the corresponding cycle type
• For the term $x_1^3$, there are $2^3 = 8$ ways to assign edges (each vertex can be connected to 0, 1, or 2 other vertices)
• For the term $x_1x_2$, there are $2^2 = 4$ ways to assign edges (the two vertices in the 2-cycle must have the same edge configuration)
• For the term $x_3$, there are $2^1 = 2$ ways to assign edges (all vertices must have the same edge configuration)
• Substituting these values into the cycle index polynomial yields $\frac{1}{6}(8 + 3 \cdot 4 + 2 \cdot 2) = 4$ non-isomorphic graphs with 3 vertices

Advanced Counting Applications of Polya's Theorem

Multiple Sets of Objects

• When dealing with multiple sets of objects, one must consider the permutation group acting on each set and compute the cycle index polynomial for the product of these groups
• For example, when counting the number of distinct ways to color the faces of a cube with two colors for the top and bottom faces and three colors for the side faces, one must consider the product of the cycle index polynomials for the permutation groups acting on the top/bottom faces and the side faces separately

Color Restrictions

• Color restrictions can be incorporated into the counting process by modifying the substitution step in Polya's Theorem
• For example, if certain colors are not allowed to be adjacent in a necklace, one must adjust the cycle index polynomial to account for these restrictions before substituting the number of colors
• This can be done by introducing additional variables to represent the allowed color configurations and modifying the terms of the cycle index polynomial accordingly

Extensions of Polya's Theorem

• The Pólya-Redfield Theorem considers the number of orbits of a group action on functions, allowing for more complex coloring schemes
• The de Bruijn's Theorem deals with the enumeration of equivalence classes of functions under permutation groups, which has applications in combinatorial design theory
• These extensions demonstrate the versatility of Polya's Theorem in solving a wide range of advanced counting problems