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A324953 Number of path change-ringing sequences of length n for 9 bells.

%I A324953 #11 May 17 2025 09:55:05
%S A324953 1,54,2862,150690,7905894,413992474,21654687592,1131904942380
%N A324953 Number of path change-ringing sequences of length n for 9 bells.
%C A324953 a(n) is the number of (change-ringing) sequences of length[*] n when we are looking at sequences of permutations of the set {1,2,3,4,5,6,7,8,9} that satisfy:
%C A324953 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 A324953 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 A324953 3. The sequence must start with the permutation (1,2,3,4,5,6,7,8,9).
%C A324953 And does not satisfy:
%C A324953 4. The sequence must end with the same permutation that it started with.
%C A324953 [*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.
%C A324953 With this [*] definition of the length of a change-ringing sequence; for 9 bells we get a maximum length of factorial(9)=362880, thus we have 362880 possible lengths, namely 1,2,...,362880. Hence {a(n)} has 362880 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 path change-ringing sequences of length n for m bells as a_m(n), {a_m(n)} has factorial(m) terms for all m.
%H A324953 Jonas K. Sønsteby, <a href="https://github.com/jonassonsteby/change-ringing">Python program</a>.
%H A324953 <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>.
%o A324953 (Python 3.7) # See Jonas K. Sønsteby link.
%Y A324953 4 bells: A324942, A324943.
%Y A324953 5 bells: A324944, A324945.
%Y A324953 6 bells: A324946, A324947.
%Y A324953 7 bells: A324948, A324949.
%Y A324953 8 bells: A324950, A324951.
%Y A324953 9 bells: A324952, This sequence.
%Y A324953 Number of allowable transition rules: A000071.
%K A324953 nonn,fini,more
%O A324953 1,2
%A A324953 _Jonas K. Sønsteby_, May 01 2019
%E A324953 a(6)-a(8) from _Bert Dobbelaere_, May 17 2025