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A324944 Number of cyclic change-ringing sequences of length n for 5 bells.

%I A324944 #28 Jul 25 2019 03:09:09
%S A324944 1,7,18,50,120,418,2114,10140,41544,164022,730136,3770982,20541820,
%T A324944 110476618,580834748,3013771544,15539996378,79715421726
%N A324944 Number of cyclic change-ringing sequences of length n for 5 bells.
%C A324944 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} that satisfy:
%C A324944 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 A324944 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 A324944 3. The sequence must start with the permutation (1,2,3,4,5).
%C A324944 4. The sequence must end with the same permutation that it started with.
%C A324944 [*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.
%C A324944 With this [*] definition of the length of a change-ringing sequence; for 5 bells we get a maximum length of factorial(5)=120, thus we have 120 possible lengths, namely 1,2,...,120. Hence {a(n)} has 120 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 A324944 Jonas K. Sønsteby, <a href="https://github.com/jonassonsteby/change-ringing">Python program</a>.
%H A324944 <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>
%o A324944 (Python 3.7) See Jonas K. Sønsteby link.
%Y A324944 4 bells: A324942, A324943.
%Y A324944 5 bells: This sequence, A324945.
%Y A324944 6 bells: A324946, A324947.
%Y A324944 7 bells: A324948, A324949.
%Y A324944 8 bells: A324950, A324951.
%Y A324944 9 bells: A324952, A324953.
%Y A324944 Number of allowable transition rules: A000071.
%K A324944 nonn,fini,more
%O A324944 1,2
%A A324944 _Jonas K. Sønsteby_, Mar 20 2019
%E A324944 a(13)-a(18) from _Bert Dobbelaere_, Jul 25 2019