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A324942 Number of cyclic change-ringing sequences of length n for 4 bells.

%I A324942 #32 Apr 29 2019 21:04:40
%S A324942 1,4,6,6,0,4,28,106,282,660,1496,3344,7176,14616,27560,47672,76092,
%T A324942 112416,148808,166960,148848,98560,43424,10792
%N A324942 Number of cyclic change-ringing sequences of length n for 4 bells.
%C A324942 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} that satisfy:
%C A324942 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 A324942 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 A324942 3. The sequence must start with the permutation (1,2,3,4).
%C A324942 4. The sequence must end with the same permutation that it started with.
%C A324942 [*]: We define the length of a change-ringing sequence to be the number of permutations in the sequence.
%C A324942 With this [*] definition of the length of a change-ringing sequence; for 4 bells we get a maximum length of factorial(4)=24, thus we have 24 possible lengths, namely 1,2,...,24. Hence {a(n)} has 24 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 A324942 Jonas K. Sønsteby, <a href="https://github.com/jonassonsteby/change-ringing">Python program</a>.
%H A324942 <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>
%o A324942 (Python 3.7) See Jonas K. Sønsteby link.
%Y A324942 4 bells: This sequence, A324943.
%Y A324942 5 bells: A324944, A324945.
%Y A324942 6 bells: A324946, A324947.
%Y A324942 7 bells: A324948, A324949.
%Y A324942 8 bells: A324950, A324951.
%Y A324942 9 bells: A324952, A324953.
%Y A324942 Number of allowable transition rules: A000071.
%K A324942 nonn,fini,full
%O A324942 1,2
%A A324942 _Jonas K. Sønsteby_, Mar 20 2019