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A324950 Number of cyclic change-ringing sequences of length n for 8 bells.

%I A324950 #9 May 17 2025 09:55:23
%S A324950 1,33,408,7360,131400,2510632,50991416,1103346172,25248402996,
%T A324950 604074338460
%N A324950 Number of cyclic change-ringing sequences of length n for 8 bells.
%C A324950 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 A324950 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 A324950 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 A324950 3. The sequence must start with the permutation (1,2,3,4,5,6,7,8).
%C A324950 4. The sequence must end with the same permutation that it started with.
%C A324950 [*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.
%C A324950 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 A324950 Jonas K. Sønsteby, <a href="https://github.com/jonassonsteby/change-ringing">Python program</a>.
%H A324950 <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>.
%o A324950 (Python 3.7) # See Jonas K. Sønsteby link.
%Y A324950 4 bells: A324942, A324943.
%Y A324950 5 bells: A324944, A324945.
%Y A324950 6 bells: A324946, A324947.
%Y A324950 7 bells: A324948, A324949.
%Y A324950 8 bells: This sequence, A324951.
%Y A324950 9 bells: A324952, A324953.
%Y A324950 Number of allowable transition rules: A000071.
%K A324950 nonn,fini,more
%O A324950 1,2
%A A324950 _Jonas K. Sønsteby_, May 01 2019
%E A324950 a(7)-a(10) from _Bert Dobbelaere_, May 17 2025