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H54 (a) Find the number of rotational orbits of seven-letter strings that use the letters A and B. (b) Find the number of ways to arrange seven beads of two colors on a circular loop that can be rotated or flipped. (c) Find examples that explain the difference between your answers to (a) and (b). H55 Repeat (a) and (b) of problem H54, using eight letters or beads.

H54 (a) Find the number of rotational orbits of seven-letter strings that use the letters A and B. (b) Find the number of ways to arrange seven beads of two colors on a circular loop that can be rotated or flipped. (c) Find examples that explain the difference between your answers to (a) and (b). H55 Repeat (a) and (b) of problem H54, using eight letters or beads.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please use the figure in the second picture to answer it.
H54 (a) Find the number of rotational orbits of seven-letter strings that use the letters A and B. (b) Find the number of ways to arrange seven beads of two colors on a circular loop that can be rotated or flipped. (c) Find examples that explain the difference between your answers to (a) and (b). H55 Repeat (a) and (b) of problem H54, using eight letters or beads.
Transcribed Image Text:H54 (a) Find the number of rotational orbits of seven-letter strings that use the letters A and B. (b) Find the number of ways to arrange seven beads of two colors on a circular loop that can be rotated or flipped. (c) Find examples that explain the difference between your answers to (a) and (b). H55 Repeat (a) and (b) of problem H54, using eight letters or beads.
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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