%I #4 Jul 27 2019 12:12:01

%S 1,33,408,7360,131400,2510632

%N Number of cyclic change-ringing sequences of length n for 8 bells.

%C a(n) is the number of (change-ringing) sequences of length[*] n when we are looking at sequences of permutations of the set {1,2,3,4,5,6,7,8} that satisfy:

%C 1. The position of each bell (number) from one permutation to the next can stay the same or move by at most one place.

%C 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.

%C 3. The sequence must start with the permutation (1,2,3,4,5,6,7,8).

%C 4. The sequence must end with the same permutation that it started with.

%C [*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.

%C With this [*] definition of the length of a change-ringing sequence; for 8 bells we get a maximum length of factorial(8)=40320, thus we have 40320 possible lengths, namely 1,2,...,40320. Hence {a(n)} has 40320 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.

%H Jonas K. Sønsteby, <a href="https://github.com/jonassonsteby/change-ringing">Python program</a>.

%H <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>.

%o (Python 3.7) See Jonas K. Sønsteby link.

%Y 4 bells: A324942, A324943.

%Y 5 bells: A324944, A324945.

%Y 6 bells: A324946, A324947.

%Y 7 bells: A324948, A324949.

%Y 8 bells: This sequence, A324951.

%Y 9 bells: A324952, A324953.

%Y Number of allowable transition rules: A000071.

%K nonn,fini,more

%O 1,2

%A _Jonas K. Sønsteby_, May 01 2019