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H60 (a) Modify the cycle index polynomial given after problem H37 to take into account reflections of the figure as well as rotations. (b) Find the number of colorings of the nine squares using color A on five squares and B on four, if the figure can be flipped as well as rotated. H37* Determine the cycle code for each rotation Ro, R₂, and R3. Next, we construct a polynomial associated with rotations of the nine- square figure by averaging the cycle codes of the four rotations. X² + 2X₁X² + X₁X²+

H60 (a) Modify the cycle index polynomial given after problem H37 to take into account reflections of the figure as well as rotations. (b) Find the number of colorings of the nine squares using color A on five squares and B on four, if the figure can be flipped as well as rotated. H37* Determine the cycle code for each rotation Ro, R₂, and R3. Next, we construct a polynomial associated with rotations of the nine- square figure by averaging the cycle codes of the four rotations. X² + 2X₁X² + X₁X²+

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.5: Polynomial And Rational Functions
Problem 26E
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Answer H60
H60 (a) Modify the cycle index polynomial given after problem H37 to take into account reflections of the figure as well as rotations. (b) Find the number of colorings of the nine squares using color A on five squares and B on four, if the figure can be flipped as well as rotated. H37* Determine the cycle code for each rotation R₁, R₂, and R3. Next, we construct a polynomial associated with rotations of the nine- square figure by averaging the cycle codes of the four rotations. X₁ + 2X₁X² + X₁ X₂ 4 This is known as the cycle index polynomial for rotations of the figure. There is also a single-variable version, obtained by setting each X; = X: X⁹ + X³ + 2X³ 4 To distinguish between these two polynomials, we refer to the latter simply as the cycle polynomial. The significance of these polynomials is demonstrated by the following problems.
Transcribed Image Text:H60 (a) Modify the cycle index polynomial given after problem H37 to take into account reflections of the figure as well as rotations. (b) Find the number of colorings of the nine squares using color A on five squares and B on four, if the figure can be flipped as well as rotated. H37* Determine the cycle code for each rotation R₁, R₂, and R3. Next, we construct a polynomial associated with rotations of the nine- square figure by averaging the cycle codes of the four rotations. X₁ + 2X₁X² + X₁ X₂ 4 This is known as the cycle index polynomial for rotations of the figure. There is also a single-variable version, obtained by setting each X; = X: X⁹ + X³ + 2X³ 4 To distinguish between these two polynomials, we refer to the latter simply as the cycle polynomial. The significance of these polynomials is demonstrated by the following problems.
Pólya-Redfield Counting Method Objects to be colored: Transformations: Extras: # transf. in group # objects to be colored: #transformations: #colors: Combined cycle codes (monomials) Check: Sum (subindex*exponent) =#objects Transformations (grouped by pattern) Pattern Check: Total number of transformations= Cycle index polynomial: (av. of cycle codes=sum of 4th column/# transformations) Cycle polynomial: (X; = X in cycle index polynomial) Pattern inventory: (For k colors: X₂ = A + A₂+ A₂+... + Ain cycle index polynomial) # k-colorings (no restrictions, each color can appear any number of times, including 0): substitute X = k in the cycle polynomial. #k-colorings when color i is used exactly m; times (m₁ + m₂ + ... + mk = # objects): find the coefficient of AAA...Ak in the pattern inventory.
Transcribed Image Text:Pólya-Redfield Counting Method Objects to be colored: Transformations: Extras: # transf. in group # objects to be colored: #transformations: #colors: Combined cycle codes (monomials) Check: Sum (subindex*exponent) =#objects Transformations (grouped by pattern) Pattern Check: Total number of transformations= Cycle index polynomial: (av. of cycle codes=sum of 4th column/# transformations) Cycle polynomial: (X; = X in cycle index polynomial) Pattern inventory: (For k colors: X₂ = A + A₂+ A₂+... + Ain cycle index polynomial) # k-colorings (no restrictions, each color can appear any number of times, including 0): substitute X = k in the cycle polynomial. #k-colorings when color i is used exactly m; times (m₁ + m₂ + ... + mk = # objects): find the coefficient of AAA...Ak in the pattern inventory.
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