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COMMENTS
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a(n) is the number of (change-ringing) sequences of length[*] n when we are looking at sequences of permutations of the set {1,2,3,4,5,6,7} that satisfy:
1. The position of each bell (number) from one permutation to the next can stay the same or move by at most one place.
2. No permutation can be repeated except for the starting permutation which can be repeated at most once at the end of the sequence to accommodate criterion 4.
3. The sequence must start with the permutation (1,2,3,4,5,6,7).
4. The sequence must end with the same permutation that it started with.
[*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.
With this [*] definition of the length of a change-ringing sequence; for 7 bells we get a maximum length of factorial(7)=5040, thus we have 5040 possible lengths, namely 1,2,...,5040. Hence {a(n)} has 5040 terms. For m bells, where m is a natural number larger than zero, we get a maximum length of factorial(m). When denoting the number of cyclic change-ringing sequences of length n for m bells as a_m(n), {a_m(n)} has factorial(m) terms for all m.
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