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.

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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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